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Posts Tagged ‘Einstein’

A major science story broke on Thursday (11th of February 2016) – the first ever detection of gravitational waves. I am currently in South America giving astronomy talks on a cruise, and therefore my internet access is very slow and very very expensive.

So, I will do a more detailed blogpost about this later next week, hopefully by Tuesday (17th). In the meantime, you can find out more about this very exciting announcement and what it means for astronomy by going to your favourite news website. From the little that I have been able to see, the coverage and explanations on the BBC’s science pages is hard to beat. 

More next week!!

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A student asked me last week if I could explain the difference between time dilation in Special Relativity (SR) and that in General Relativity (GR), so here is my attempt at doing so. Time dilation in SR comes about when something travels near the speed of light, and is due to the Lorentz transformations which ensure that experiments in any inertial frame are indistinguishable from each other.

I have already derived the Lorentz Transformations from first principles in this blog, and these equations are at the heart of SR, and show why time dilation occurs when one travels near the speed of light. In this blog here, I worked through some examples of time dilation in SR. But, what about time dilation in GR?

How does time dilation come about in GR?

As I have already explained in this blog here, Einstein’s principle of equivalence tells us that whatever is true for acceleration is true for a gravitational field. So, to see how gravity affects time the easiest way is to consider how time would be affected in an accelerating rocket.

We will consider a rocket in empty space, away from any gravitational fields, which is accelerating with an acceleration g. We will have two people in the rocket, Alice and Bob. Alice is at the top end of the rocket, the nose end. Bob is at the bottom end of the rocket, where the tail is. Alice sends two pulses of light, one at time t=0, and the second one at a time t= \Delta \tau_{A} later. They are received at the back of the rocket by Bob; the first pulse is received when the time is t=t_{1}, and the second one when the time is t=t_{2} = t_{1} + \Delta \tau_{B}, where \Delta \tau_{B} is the time interval between flashes as measured by Bob.

This is illustrated in the figure below.

We can see how time dilation comes about in GR by considering a rocket accelerating in space with an acceleration g, and a light flashing at the top of the rocket and being received at the bottom

We can see how time dilation comes about in GR by considering a rocket accelerating in empty space with an acceleration g, and a light flashing from Alice at the front-end of the rocket and being received at the back-end by Bob.

We will set it up so that Bob’s position at time t=0 when the first flash is emitted by Alice is z_{B}(0)=0, and so his position at any other time is given by
z_{B}(t) = \frac{1}{2}gt^{2} \text{ (Equ. 1) }

(this just comes from Newton’s 2nd equation of motion s=ut + \frac{1}{2}at^{2}, see my blog here which derives those equations).

The position of Alice will just be Bob’s position plus the distance between them, which we will call h (the height of the rocket), so
z_{A}(t) = h + \frac{1}{2}gt^{2} \text{ (Equ. 2) }

We will assume that the first pulse takes a time of t=t_{1} to travel from Alice to Bob. The second pulse is emitted by Alice at a time \Delta \tau_{A} after the first pulse, this is the time interval between each light pulse that Alice sends. This second pulse is received by Bob at a time of t_{2} = t_{1} + \Delta \tau_{B}, where \Delta \tau_{B} is the time interval between pulses as measured by Bob using a clock next to him.

When the first pulse leaves Alice her position is z_{A}(0), which from equation (2) is h, as she is at the top of the rocket. When Bob receives the pulse at time t=t_{1} his position will be z_{B}(t_{1}) which, from equation (1) is z_{B}(t_{1}) = \frac{1}{2} gt_{1}^{2}. So, the distance travelled by the pulse is going to be
z_{A}(0) - z_{B}(t_{1})= ct_{1} \text{ (Equ. 3) }

as the speed of light is c and it travels for t_{1} seconds. Because the rocket is accelerating, the distance travelled by the second pulse will not be same (as it would be if the rocket were moving with a constant velocity). The distance travelled by the second pulse will be less, and is given by
z_{A}(\Delta \tau_{A}) - z_{B}(t_{1} + \Delta \tau_{B}) = c(t_{1} + \Delta \tau_{B} - \Delta \tau_{A}) \text{ (Equ. 4) }

We can use Equations (1) and (2), which give expressions for z_{B} \text{ and } z_{A} as a function of t, to put in the values that z_{A} \text{ and } z_{B} would have when t = \Delta \tau_{A} for z_{A} and (t_{1} + \Delta \tau_{B}) for z_{B} respectively.

Substituting from Equations (1) and (2) into Equation (3) we have
z_{A}(0) = h, \; \; z_{B}(t_{1}) = \frac{1}{2}gt_{1}^{2}

which makes equation (3) become
h - \frac{1}{2}gt_{1}^{2} = ct_{1} \text{ (Equ. 5) }

Doing the same kind of substitution into equation (4) we have
z_{A}(\Delta \tau_{A}) = h + \frac{1}{2}g \left( \Delta \tau_{A} \right)^{2} \rightarrow h

z_{B}(t_{1} + \Delta \tau_{B}) = \frac{1}{2}g(t_{1} + \Delta \tau_{B})^{2} \rightarrow \frac{1}{2}gt_{1}^2 +gt_{1} \Delta \tau_{B}

assuming that we can ignore terms in (\Delta \tau_{A})^{2} \text{ and } (\Delta \tau_{B})^{2}

Substituting these expressions into equation (4) gives
h - \frac{1}{2}gt_{1}^{2} - gt_{1} \Delta \tau_{B} = c(t_{1} + \Delta \tau_{B} - \Delta \tau_{A}) \text{ (Equ. 6) }

We now subtract equation (6) from (5) to give
gt_{1} \Delta \tau_{B} = c \Delta \tau_{A} - c \Delta \tau_{B} \text{ Equ. (7) }

Re-arranging equation (5) as \frac{1}{2}gt_{1}^{2} +ct^{1} -h and using the quadratic formula to find t_{1} we can write that
t_{1} = \frac{ -c \pm \sqrt{ c^{2} + 2gh } }{ g } \rightarrow \frac{ -c + \sqrt{ c^{2} + 2gh } }{ g }

(we can ignore the negative solution because the time is always positive). We will next use the binomial expansion to write
\sqrt{ c^{2} + 2gh } \approx c ( 1 + \frac{gh}{ c^{2} } )

(where we have ignored terms in \left( \frac{2gh}{c^{2}} \right)^{2} and higher in the Binomial expansion), and so we can write for t_{1}
t_{1} \approx \left( \frac{ -c + c \left( 1 + \frac{gh}{ c^{2} } \right) }{ g } \right) \rightarrow gt_{1} = c \left( \frac{gh}{ c^{2} } \right)

Substituting this expression for gt_{1} into equation (7) we now have
c \left( \frac{gh}{ c^{2} } \right) \Delta \tau_{B} = c \Delta \tau_{A} - c \Delta \tau_{B}

We can cancel the c in each term and bringing the terms in \Delta \tau_{B} onto one side and the term in \Delta \tau_{A} on the other side we have
\Delta \tau_{B} \left( 1 + \frac{ gh }{ c^{2} } \right) = \Delta \tau_{A}

and so
\Delta \tau_{B} = \frac{ \Delta \tau_{A} }{ \left( 1 + \frac{ gh }{ c^{2} } \right) }

and using the binomial expansion for (1 + gh/c^{2})^{-1} (and ignoring terms in \left( \frac{ gh }{ c^{2} } \right)^{2} and higher), we can finally write
\boxed{ \Delta \tau_{B} = \Delta \tau_{A} \left( 1 - \frac{ gh }{ c^{2} } \right) \text{ (Equ. 8) } }

Because \frac{gh}{c^{2}} is always positive, this means that \Delta \tau_{B} is always less than \Delta \tau_{A}, or to put it another way the time interval as measured by Bob at the back-end of the rocket will always be less than the time interval measured by Alice where the light pulses were sent. This means that Bob will measure time to be going at a slower rate than Alice, Bob’s time will be dilated compared to Alice.

From the principle of equivalence, whatever is true for acceleration is true for gravity, so if we now imagine the rocket stationary on the Earth’s surface, with the top end in a weaker gravitational field than the bottom end, we can see that a gravitational field will also lead to pulses arriving at Bob being measured closer together than where they were emitted by Alice. So, gravity slows clocks down!

A very important difference between time dilation in SR and time dilation in GR is that the time dilation in GR is not symmetrical. In SR, both observers in their respective inertial frames think it is the other person’s clock which is running slow. In GR, both Alice and Bob will agree that it is Bob’s clock which is running slower than Alice’s clock.

In a future blog I will do some calculations on this effect in different situations, but as you can see from Equation (8), the size of the dilation depends on the acceleration g and the difference in height between A \text{ and } B. I will also discuss whether it is time dilation due to GR or time dilation due to SR which affect the satellites which give us GPS the more, as both effects have to be taken into account to get the accuracy we seek in the GPS position.

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To most of us, inflation is a nasty thing which sees the money in our pocket being worth less as prices go up. It’s a bad thing! But, in cosmology, a theory called cosmic inflation explains very neatly several key properties of the Universe. The theory of cosmic inflation was first suggested by Alan Guth in 1980, and yesterday (Monday the 17th of March 2014) a team led by John Kovak of Harvard University announced the first direct evidence that cosmic inflation did actually happen. There is also a Cardiff University involvement in this project.



The story on the confirmation of cosmic inflation as it appeared on the BBC science website.

The story on the confirmation of cosmic inflation as it appeared on the BBC science website.



What is cosmic inflation?

In 1980, particle physicist Alan Guth was pondering some of the observed properties of the Universe, and he came up with the idea of cosmic inflation. The observed properties he was hoping to explain with his theory were

  • the “Horizon problem”
  • the “Flatness problem”
  • the “Magnetic-monopole problem”

The Horizon problem

When the cosmic microwave background radiation (CMBR) (the prediction of which I blogged about here) was discovered in 1964 it was recognised that it was most probably the “echo” of the Big Bang. By 1967 Bruce Partridge and David Wilkinson of Princeton University showed that the CMBR was the same from all parts of the sky down to a level of 0.1% of its 3 Kelvin temperature.

It was realised soon after this that this presented a problem, the so called “horizon problem”. It is actually perplexing that different parts of the sky should have the same CMBR temperature because when we look in different parts of the sky we are looking at parts of space which have not had the time to be in contact with each other in any way; they are simply too far apart. Therefore, a patch of sky in one direction with a particular CMBR temperature should have no knowledge of the CMBR temperature of a patch of sky in a different direction.

This is a little bit like switching on a heater in the centre of a large room. Everyone knows that it will take time for the whole room to come to the same temperature, and if the room were really really big you would not expect the corners which are far away from the heater to have the same temperature as the centre of the room next to the heater after just a few minutes. The heat just hasn’t had enough time to spread throughout the room. So, if you found that the whole room was at the same temperature, even though the heat hadn’t had enough time to spread throughout the room, it would be a bit of a puzzle. That is, in essence, the “horizon problem”.

The flatness problem

Einstein showed in his theory of gravity, the General Theory of Relativity, that gravity causes space to bend. A Universe with lots of matter in it will have a different geometry (shape) to a Universe with less matter in it. The so-called “critical density” of the Universe would be a density that would give it a flat geometry. It was realised since the 1960s that the density of the Universe seemed to be very close to the critical density. Why should this be, when it could have any value. It could be much much more or much much less? If you do the mathematics, for the density to be within about a factor of two of the critical density today means it had to have been incredibly close to the critical density in the earliest moments of the Universe. Close to about one part in 10^{60}!! This is the “flatness problem”.

The magnetic monopole problem

In electricity, we are all familiar with positive and negative charges. James Clerk Maxwell showed in the mid 1800s that electricity and magnetism are part of the same force, electromagnetism. And yet, you never find a magnetic monopole, you always find magnetic poles come in pairs, they always have both a north and south pole. Theoretically there is no reason why one shouldn’t find just e.g. a north pole on its own, without a south pole. This is the “magnetic monopole problem”.

What is cosmic inflation?

Alan Guth’s idea of cosmic inflation suggested that when the Universe was incredibly young, some 10^{-36} seconds old, it went through a brief period of very rapid expansion. This period ended when the Universe was about 10^{-33} \text{ or } 10^{-32} seconds old, but in this incredibly brief period Guth argued that the Universe grew from being much smaller than a proton to something about the size of a marble. After this brief period of very rapid expansion (inflation), the expansion of the Universe settled down to the more sedate rate of expansion that we see today.

How does cosmic inflation solve these three problems?

The horizon problem is solved by inflation because the very rapid expansion which inflation proposes would allow parts of the Universe which are now too far apart to have ever communicated with each other to have been close enough together before inflation. So, going back to my analogy with the room being heated, it is as if the room started off really small, so small that all parts of it could come to the same temperature, then it suddenly expanded so that the room we are now looking at is much much bigger.

The flatness problem is solved by cosmic inflation by drawing the analogy between the geometry of the Universe and a curved surface. If a curved surface is large enough, then on a local scale it is always going to look flat. An easy analogy to understand this is the surface of our Earth. We all know it is spherical, but on a local scale it appears flat. If the Universe underwent a period of cosmic inflation, then we are seeing such a small part of it that the small part we see is always going to appear flat, no matter what the overall geometry.

The magnetic monopole problem is solved by cosmic inflation in the following manner. The idea is that magnetic monopoles were created in large quantities before the period of cosmic inflation. They should still exist today, but because the Universe expanded so rapidly during cosmic inflation, their number density (how many there are per unit volume) is so tiny that we haven’t found any in the part of the Universe which we are able to observe.

The discovery made by BICEP2

Until yesterday, there had been no direct evidence of anything that cosmic inflation predicted, only agreement between the theory and things which had already been observed. One prediction of the theory is that the CMBR should be polarised in a particular way with a particular amount of polarisation (you can think of polarisation as a particular twisting of radiation, instead of vibrating in all directions it only vibrates in particular directions). The BICEP2 experiment (“Background Imaging of Cosmic Extragalactic Polarization”, the “2” indicates it is the second generation of this experiment) has been using the South Pole Telescope which is, as the name implies, at the Earth’s south pole, and has been looking for a particular signature in the CMBR – the “B-mode polarisation” as it is called.

Yesterday the team announced that they had, for the first time, detected this B-mode polarisation, which is the most direct evidence yet that the theory of cosmic inflation is correct. This polarisation comes about due to gravitational waves in the very very early Universe, so the detection of the B-mode polarisation is also direct evidence of gravitational waves, which were predicted by Einstein but have never been directly detected before.

If you want to read the actual announcement paper you can find the pre-print by following this link here. Here is a screen capture of the first page of the paper.



The first page of the paper announcing the detection of evidence for cosmic inflation.

The first page of the paper announcing the detection of evidence for cosmic inflation. Notice that Cardiff University has an involvement with Peter Ade being the first author in the alphabetical list.



Superimposed on the variations in the temperature of the cosmic microwave background (red and blue blobs) is the evidence for the B-mode polarisation (the small swirls or black lines).

Superimposed on the variations in the temperature of the cosmic microwave background (red and blue blobs) is the evidence for the B-mode polarisation (the small black swirls).



This is very exciting news for cosmology and our understanding of the earliest moments of the Universe. It suggests that our model of the early Universe, including the theory of cosmic inflation, is correct (or at least is on the right tracks). Little by little, astronomers are unfolding the mysteries of the very earliest moments of creation!

If you want to read a more technical (but still non-specialist) explanation, then this story in Sky & Telescope is pretty good. Or, you may prefer this from Sean Carroll’s blog.

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The European Space Agency announced last week (28th of November) that a space-based gravitational wave observatory will form one of its next two large science missions. This is exciting news for Cardiff University, as it has a very active gravitational waves research group headed up by Professor B S Sathyaprakash (known to everyone as “Sathya”). The Cardiff group, along with others in the Disunited Kingdom, played an important role in persuading ESA to make this observatory one of its next large science missions. Just over a year ago I blogged about some theoretical modelling of gravitational waves the Cardiff group had done.

The ESA plan is to launch a space-based gravitational wave observatory in 2034, which will have much more sensitivity than any current or even future ground-based gravitational wave observatory. Although NASA also had plans to launch a space-based gravitational wave observatory, there is currently none in existence, so this is really ground-breaking technology that ESA is announcing. From what I can understand, the current announcement by ESA is their commitment to the proposed LISA (Laser Interferometer Space Antenna) gravitational wave observatory. It would seem NASA has withdrawn their commitment to what was originally going to be a joint NASA/ESA mission.



ESA has chosen a gravitational wave detector to be funded.

ESA has chosen a space-based gravitational wave detector to be funded as one of its two key future missions. It should go into operation in 2034.



What are gravitational waves?

Gravitational waves are ripples in the fabric of space. They were predicted by Einstein as part of his theory of general relativity, the best theory we currently have to describe gravity. In his theory, events which involve extreme gravitational forces (such as two neutron stars orbiting each other (or merging), or the creation of a black hole) will lead to the emission of these gravitational waves. As they spread out from the source at the speed of light, they literally deform space as they pass by, just as ripples deform the surface of a pond as they spread out from a dropped stone.



An artist's impression of gravitational waves being produced around two orbiting neutron stars.

An artist’s impression of gravitational waves being produced as two black holes orbit each other.



Current gravitational wave observatories

There are several current gravitational wave observatories, all ground-based. These include VIRGO (in Italy) and LIGO (Laser Interferometer Gravitational Wave Observatory) which is in the United States. LIGO is the most sensitive of the current generation of gravitational wave detectors. LIGO actually comprises three separate detectors; one in Livingston, Louisiana and two in Hanford, Washington State. Each of the three separate detectors consists of two long arms at right angles to each other, forming a letter “L”. The idea behind these detectors is that, if a gravitational wave were to pass the detector, each of the two arms would have its length changed differently by the deformation of space as the gravitational wave passes through. Thus the detectors work on the principle of an interferometer, looking for tiny changes in the relative length of the two arms. And, when I say tiny, I mean tiny. In a 4km arm they are looking for changes of the order of 10^{-18} \text{ m}, or about one thousandth the size of a proton!



The principle of a gravitational wave detector.

The principle of a gravitational wave detector. They are essentially “interferometers”, with two arms at right angles to each other. As the gravitational waves pass the detector, space will be deformed and alter the relative lengths of the two arms.



LIGO (Laser Interferometer Gravitational Wave Detector)

Currently the most sensitive gravitational wave detector is LIGO. LIGO consists of three separate detectors, one in Livingston, Louisiana and two in Hanford, Washington State. The detector in Louisiana is shown below.



The LIGO detector in Livingstone, Louisiana. Each arm is 4km in length, and can detect changes in the relative length of the two arms of less than the size of a proton.

The LIGO detector in Livingstone, Louisiana. Each arm is 4km in length, and can detect changes in the relative length of the two arms of less than the size of a proton.



The detector in Louisiana, and one of the two detectors in Washington State, consist of two 4km long arms at right angles to each other. An event like the collapse of a 10 solar-mass star into a black hole is expected to produce a change in length in a 4km arm of about 10^{-18} \text{ m}, which is about one thousandth the size of a proton. This is just at the limit of the detection capabilities of LIGO, which is why astrophysicists are wanting more sensitive detectors to be placed into space. The other detector in Washington State has arms which are 2km in length, but just as sensitive as the detector with 4km arms at frequencies above 200 Hz, due to a different design.

LISA – Laser Interferometer Space Antenna

The ESA plans just announced will be based on the NASA/ESA plans for LISA, which have been on the drawing board for most of the last 10 years. ESA’s plan is to build two space-based interferometers, which will be in the form of equilateral triangles as this artist’s description shows.



An artist's impression of LISA, the "Laser Interferometer Space Antenna". Each interferometer will consist of an equilateral triangle, with each side 5 million km in length.

An artist’s impression of LISA, the “Laser Interferometer Space Antenna”. Each interferometer will consist of an equilateral triangle, with each side 5 million km in length.

The plan for LISA is to have arms which are 5 million km long! Compare this to the 4km long arms of LIGO. The changes in the length of a 5 million km long arm would be roughly one million times more than for LIGO, so rather than 10^{-18} \text{ m it would be } 10^{-12} \text{ m}, which should be well within the capabilities of the detectors. This means that less energetic events than the collapse of a 10-solar mass star into a black hole would be detectable by LISA. All kinds of astrophysical events which involve large changes in gravitational fields should be detectable by LISA, including the afore-mentioned creation of black holes, but also the merging of neutron stars, and even the merging of less massive stars.

But, possibly most exciting is the opportunity that gravitational waves provide to probe the very earliest moments after the Big Bang. With normal electromagnetic radiation (light, x-rays, infrared light etc.), we can only see as far back as about 300,000 years after the Big Bang. This is when the Cosmic Microwave Background Radiation was produced. Prior to this time, the Universe was opaque to EM radiation of any wavelength, because it was full of unbound electrons, and the photons would just scatter off of them and not get anywhere (see my blog here about the CMB). But, it was not opaque to gravitational waves, so they provide a way for us to see back beyond the CMB, and a unique way to learn about the conditions of the Universe in its earliest moments.

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I should say right at the start, Richard Feynman is my scientific hero. He was one of the most naturally gifted communicators of complex topics and ideas that I can think of, a born communicator of his subject. I first came across him when I was 17, seeing him in the now famous BBC Horizon interview “The Pleasure of finding things out”.

 

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In fact, a recent BBC Horizon programme, “The fantastic Mr. Feynman” referred to him as “the greatest communicator of science the World has ever seen”. Quite an accolade! I will do a separate blog about this programme and some of the interesting points about Feynman which it raised in the near future.

Feynman is in this list for his work on Quantum Electro Dymanics (QED), the theory we have which describes the behaviour of electrons. In 1947/48 Feynman produced a method to be able to make sensible calculations from the Quantum Mechanical theory of the motion of electrons which had been laid down in the 1920s by Schrödinger, Heisenberg and Dirac. In 1965 he won the Nobel Prize in Physics in recognition of this work.

He is generally recognised as one of the towering intellects of 20th Century physics. Because he was such a fun and crazy character, it is sometimes easy to overlook his important contributions to 20th Century physics. Instead we get lost in the stories of cracking safes at Los Alamos, or playing bongos in the Rio de Janeiro carnival or sketching strippers in topless clubs in Pasadena.

Hans Bethe, who won a Nobel Prize himself in 1967 for his work on the triple-alpha process within stars, referred to Feynman as “a magician”. This is the quote with which he introduces Feynman on the back cover of Feynman’s autobiography

 

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And this is a quote from Brian Clegg’s book “Light Years”, in his introduction to the last chapter on Quantum Electro Dynamics (QED), the theory for which Feynman won his Nobel Prize.

 

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Feynman’s brief biography

Feynman was born in Far Rockaway, New York, in 1918. His father was a military uniform salesman. Feynman showed early brilliance in mathematics, and went to the Massachusets Institute of Technology (MIT) to major in Physics. From MIT he went to Princeton to work on his PhD. He entered Princeton with an unprecedented full marks in the Physics and Mathematics Graduate Entrance exams. Whilst finishing his PhD thesis he was invited to join the Manhattan Project in Los Alamos, to help develop the atomic bomb. He took charge of the “computing” team, manual calculations which were necessary to develop the bomb and predict its power etc. He got the computing team to work on problems in parallel, so the team went from making 3 complex calculations a year to 3 every month, a factor of 10 improvement.

After the war, Feynman took a position as Professor at Cornell University. He worked with Hans Bethe, with whom he had worked at Los Alamos. It was at Cornell that he did the work which eventually won him the Nobel Prize. In 1950 he went on a sabbatical to Brazil, and never returned to Cornell, instead taking up a position at the California Institute of Technology (Caltech), where he remained for the rest of his life. He died in 1988 of cancer, but thankfully not before he had shared many of his crazy antics in life in his best selling autobiography “Surely You’re Joking Mr. Feynman”

In this clip, which is the start of the BBC Horizon interview called “The pleasure of finding things out”, Feynman talks about how a scientist can, in some sense, gain a deeper appreciation of a flower than a non-scientist, because a scientist can appreciate the complexity of the flower at a deeper level than just its aesthetic beauty, and hence appreciate the flower at multiple levels.

 

 

The entire BBC Horizon interview “The pleasure of finding things out” is available via YouTube, if you do a search for it.

Does Feynman deserve to be in this list? If so, would he get into a “top five”?

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You can read more about Richard Feynman and the other physicists in this “10 best” list in our book 10 Physicists Who Transformed Our Understanding of the UniverseClick 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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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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In an interesting exercise, The Guardian newspaper recently drew up a list of the “10 best physicists”. I don’t think the list they compiled is in any particular order, but here it is.

 

  1. Isaac Newton (1643-1727)
  2. Niels Bohr (1885-1962)
  3. Galileo Galilei (1564-1642)
  4. Albert Einstein (1879-1955)
  5. James Clerk Maxwell (1831-1879)
  6. Michael Faraday (1791-1867)
  7. Marie Curie (1867-1934)
  8. Richard Feynman (1918-1988)
  9. Ernest Rutherford (1871-1937)
  10. Paul Dirac (1902-1984)

 

How many of these names do you recognise? Whilst some are “household names”, others are maybe only known to physicists.

Over the next several months I will post a blog about each of these entries, giving more details of what their contribution(s) to physics were. Any such list is, of course, bound to promote discussion and disagreement, and I can also see that “The Guardian” have also allowed readers to nominate their own names.

 

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In this blog I derived, from first principles, the Lorentz transformations which are used in Einstein’s special theory of relativity to relate one frame of reference S to another frame of reference S^{\prime} which are moving relative to each other with a speed v.

\boxed {\begin{array}{lcl} x^{\prime} & = & \gamma (x - vt) \\ y^{\prime} & = & y \\ z^{\prime} & = & z \\ t^{\prime} & = & \gamma ( t - \frac{ v }{ c^{2} }x ) \end{array} }

So, these relate the length x and time t in two different reference frames which are moving relative to each other with a velocity v. One of the most intriguing and surprising consequences of Einstein’s special theory of relativity is that time is relative and not absolute. What this means in simple terms is that two observers in two reference frames S and S^{\prime} moving relative to each other with a velocity v will measure time to be passing at different rates.

Time dilation

This phenomenon is known as time dilation. Let us consider our two reference frames S and S^{\prime}. We will have a clock in frame S^{\prime}, which in that reference frame is stationary (e.g. a clock on a rocket ship, although the rocket ship is moving, the clock is stationary relative to the rocket ship).

Two successive events on the clock in S^{\prime} are separated by a time interval \Delta t^{\prime} which we are going to call the proper time T_{0}. The time interval in the other reference frame, S, is \Delta t = T. How does this compare to T_{0}?.

In the reference frame S^{\prime} the clock is stationary, so we can say that the location of the clock in the x-dimension, x^{\prime}, does not change. That is, \Delta x^{\prime} = 0.

Using our equation which relates t \; \text{and} \; t^{\prime} from above, we can write

\begin{array}{lcl} \Delta t & = & \gamma (\Delta t^{\prime} + \frac{v}{c^{2}} \Delta x^{\prime}) \\ \Delta t & = & \gamma \Delta t^{\prime} \; \; (\text{as} \; \Delta x^{\prime} = 0 ) \\ \end{array}

and so we can write

\boxed {T = \gamma T_{0}}

This means the time interval T in frame S will appear to be dilated by a factor of \gamma compared to the proper time interval T_{0}.


A clock travelling at close to the speed of light will run ore slowly compared to a stationary clock

A clock travelling at close to the speed of light will run more slowly compared to a stationary clock


Time dilation in Nature

We observe the effects of time dilation every day in Nature. Cosmic rays, high energy particles from space, strike molecules in our atmosphere and create particles from the high energy interactions (this is the same as happens in the Large Hadron Collider). One of the particles created in these reactions are muons, which decay very rapidly in about 2 microseconds second (2 millionths of a second). Given the distance between where they are created in the upper atmosphere and the Earth’s surface, they should not survive long enough to make it to the surface of the Earth. But they do. How? Because of time dilation, the muons are moving so quickly that \gamma is appreciable more than 1, meaning that 2 microseconds in the muon’s frame of reference is much longer in our frame of reference. So, in the muon’s frame of reference it is indeed decaying in let us say 2 microseconds, but in our frame or reference it could survive for maybe a millisecond (thousandth of a second) or more, long enough to reach the surface of the Earth.

The symmetry of relativity

One aspect of relativity which confuses a lot of people is that it is symmetrical. Although an observer in frame S will think that the clock in frame S^{\prime} is ticking more slowly, if an observer in S^{\prime} were to look at a clock which was at rest in frame S, that observer would think that the clock in frame S is moving more slowly. Each would think that their clock is behaving normally, and it is the clock in the other’s reference frame which is showing the effects of time dilation.

The twin paradox

If a twin sets off on a space trip where the rocket will travel close to the speed of light, then time dilation effects will come into play. This means that e.g. a 20-year old twin can set off on a space trip which for the twin who stays on Earth appears to last for 40 years, but because of time dilation effects maybe only 5 years will appear to pass for the twin on the rocket. Thus, the 60-year old twin who stayed on Earth will be greeted after 40 years by a 25-year old twin!!

In the example I have shown, 40 years for the twin who stays on Earth appears to pass as 5 years for the twin on the rocket. This means the time dilation factor is 40/5 = 8, and as the time dilation factor is just the Lorentz factor \gamma, this means the rocket will need to travel at a speed of 99.2\% of the speed of light.

HANG ON!!! you say, what about the symmetry of relativity? Surely the twin in the rocket will think that the twin on Earth is aging more slowly, so why doesn’t he return to find the twin on Earth is only 25 and he is 60? Or maybe, because of the symmetry, they will both be 60 when the travelling twin returns?

No, what one has to realise is that there is no symmetry in this trip. In order for the travelling twin to leave the Earth and travel at close to the speed of light he has to speed up considerably. Also, in order to come back he has to slow down and reverse his direction, speeding up again once he’s turned his rocket around to come back to Earth. And, as he approaches Earth, he will have to slow down again. These large accelerations (changes in speed) which the travelling twin experiences break the symmetry, and so it really is the case that the travelling twin will return younger than the twin who has stayed on Earth. How much younger depends on how close to the speed of light the travelling twin travels.

Back to the future

Although it is possible therefore to “travel to the future”, as our twin in the example above does, what is not possible is to travel to the past. In order to do this one would need to travel faster than the speed of light, which Einstein’s theory does not allow. The results of neutrinos travelling faster than the speed of light, announced back in the Autumn of 2011, proved to be incorrect. One of the reasons that story caused so much interest is that travelling back in time has all kinds of problems associated with it, the movie “Back to the future” illustrated some of them. I will discuss time travel more in another blog.

Time for a photon

I will finish this blog with a question about photons (particles of light). Remember that Einstein’s theory of special relativity is based on the premise that light always travels at the same speed in a vacuum. The nearest star system beyond our Solar System is the Proxima Centauri system, which is 4.2 light years away. That means it takes light 4.2 years to travel from this system to us, which in terms of kilometres is 40 trillion kilometres (4 \times 10^{13} kilometres!). Now you know why we use light years for such large distances.

So if light takes 4.2 years to travel the 40 trillion kilometres from Proxima Centauri to Earth, my question to you is


how long would it seem to take if you were a photon moving at the speed of light?


Answers on a postcard, or in the comment section below.

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Riding on a beam of light

In this previous blog, I discussed how an experiment involving electrodynamics was not invariant under a Galilean transformation. Or, to put it another way, the laws of electrodynamics as stated would allow someone to determine whether they were at rest or moving, something which deeply troubled a young Albert Einstein. It is said that one of Einstein’s first “thought experiments” was to imagine himself travelling along on a beam of light. Light is the ultimate “free lunch”, the changing magnetic field produces a changing electric field which produces a changing magnetic field. It self-propogates at a speed of 3 \times 10^{8} metres per second in a vacuum.

Einstein realised that if he were travelling with the beam of light then, relative to him, the light would disappear as the electric and magnetic fields would be stationary relative to him. This worried him, as it suggested that one would be able to tell whether one was travelling or at rest, just by measuring the properties of light. Einstein realised, in an insight which possibly no one else was capable of, that the speed of light was fundamental to physics, and needed to always be constant. This led him to develop what we now call the special theory of relativity, most of which is expressed in a paper he published in 1905 called “On The Electrodynamics of moving bodies“.

Einstein’s special theory of relativity

Einstein’s Special Theory of Relativity is based on two very simple but far reaching principles

  1. No experiment, mechanical or electrodynamical, can distinguish between being at rest or moving at a constant velocity.
  2. That the speed of light in a vacuum, c, is constant to any observer, no matter how quickly the observer is moving.

From the second of these principles, with a simple thought experiment, we can derive the Lorentz transformations from first principles. These are the equations which allow us to translate from one frame of reference to another so that all the laws of Physics are invariant.

An expanding sphere of light

The thought experiment we will use to derive the Lorentz transformations from first principles is one of a flash of light originating at the origin of two frames of reference S and S’ which are moving relative to each other with a velocity v. We set up our experiment so that at time t=0 the origins of the two frames of reference are in the same place.

 

Two frames of reference S and S' moving relative to each other have a flash of light originate at their respective origins at time t=0

Two frames of reference S and S’ moving relative to each other with a velocity v have a flash of light originate at their respective origins at time t=0

 

The flash of light will expand as a sphere, moving with a velocity c in both frames of reference, in accordance with Einstein’s 2nd principle of relativity. For reference frame S we can write that the square of the radius r^{2} of the sphere is x^{2} + y^{2} + z^{2} = c^{2}t^{2} so

\boxed{ x^{2} + y^{2} + z^{2} - c^{2}t^{2}=0 } \qquad(1)

For the reference frame S’ we can write that

\boxed{ x^{\prime 2} + y^{\prime 2} + z^{\prime 2} - c^{2}t^{\prime 2} = 0 } \qquad(2)

These two equations must be equal, as it is the same sphere of light and therefore the sphere must have the same radius in the two reference frames. Let us see if we can transform from one to the other using the Galilean transforms, which are

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

x^{\prime 2} + y^{\prime 2} + z^{\prime 2} -c^{2}t^{2} = (x-vt)^{2} + y^{2} + z^{2} - c^{2}t^{2}

Expanding the brackets of the right hand side gives

x^{2} - 2vtx + v^{2}t^{2} + y^{2} + z^{2} - c^{2}t^{2} \neq x^{2} + y^{2} + z^{2} - c^{2}t^{2}

 

The the equation should be equal, but the terms highlighted do not exist on the right hand side of the equation.

The left side of the equation should be equal to the right side, but the terms highlighted do not exist on the right hand side of the equation.

 

As we can see, the two expressions are not equal as the left hand side has the extra terms -2vtx + v^{2}t^{2}. This means that a Galilean transformations does not work. The extra terms involve a combination of x and t, which suggests that both the equations linking x and x^{\prime} and t and t^{\prime} need to be modified, not just the equation for x as is the case in the Galilean transformations.

Modifying the Galilean transformations

Let us assume that the transformations can be written as

\boxed {\begin{array}{lcl} x^{\prime} & = & a_{1}x + a_{2}t \qquad(3) \\ y^{\prime} & = & y \\ z^{\prime} & = & z \\ t^{\prime} & = & b_{1}x + b_{2}t \qquad(4) \end{array} }

We need to find the values of a_{1}, a_{2}, b_{1} and b_{2} which correctly transform the equations for the expanding sphere of light. We do this by substituting equations (3) and (4) into equation (2). Before we do this, we note that the origin of the primed frame x^{\prime}=0 is a point that moves with speed v as seen in the unprimed frame S. Therefore its location in the unprimed frame S at time t is just x=vt. So we can write equation (3) as

x^{\prime} = 0 = a_{1}x + a_{2}t \rightarrow x = -\frac{a_{2}}{a_{1}} t = vt

\therefore \frac{ a_{2} }{ a_{1} } = -v

Re-writing equation (3)

x^{\prime} = a_{1}x + a_{2}t = a_{1}(x+\frac{ a_{2} }{ a_{1} } t) = a_{1}(x-vt)

Now we substitute this expression and equation (4) into equation (2)

a_{1}^{2}(x-vt)^{2} + y^{\prime 2} + z^{\prime 2} -c^{2}(b_{1}x+b_{2}t)^{2} = x^{2} + y^{2} + z^{2} -c^{2}t^{2}

a_{1}^{2} x^{2} -2a_{1}^{2} xvt + a_{1}^{2} v^{2} t^{2} - c^{2} b_{1}^{2} x^{2} - 2c^{2} b_{1} b_{2} xt -c^{2} b_{2}^{2} t^{2} = x^{2} - c^{2} t^{2}

Equating coefficients:

( a_{1}^{2} - c^{2}b_{1}^{2} ) x^{2} = x^{2} \rm{\;\; or \;\;} a_{1}^{2} - c^{2}b_{1}^{2} = 1 \qquad(5)

( a_{1}^{2} v^{2} - c^{2} b_{2}^{2} ) t^{2} = -c^{2} t^{2} \rm{\;\; or \;\;} c^{2} b_{2}^{2} -a_{1}^{2} v^{2} = c^{2} \qquad(6)

(2a_{1}^{2} v + 2b_{1} b_{2} c^{2} ) xt = 0 \rm{\;\; or \;\;} b_{1} b_{2} c^{2} = -a_{1}^{2}v \qquad(7)

From equations (5) and (6) we can write

b_{1}^{2} c^{2} = a_{1}^{2} - 1 \qquad(8)

and

b_{2}^{2} c^{2} = c^{2} + a_{1}^{2} v^{2} \qquad (9)

Multiplying equations (8) and (9) and squaring equation (7) we get

b_{1}^{2} b_{2}^{2} c^{4} = ( a_{1}^{2} - 1 )( c^{2} + a_{1}^{2} v^{2} ) = a_{1}^{4} v^{2}

so

a_{1}^{2} c^{2} - c^{2} + a^{4} v^{2} - a_{1}^{2} v^{2} = a_{1}^{4} v^{2}

a_{1}^{2} c^{2} - a_{1}^{2} v^{2} = c^{2}

a_{1}^{2} ( c^{2} - v^{2} ) = c^{2}

a_{1}^{2} = \frac{ c^{2} }{ c^{2} - v^{2} } = \frac{ 1 }{ 1 - v^{2}/c^{2} }

so

\boxed{ a_{1} = \frac{ 1 }{ \sqrt{ (1 - v^{2}/c^{2} ) } } }

Thus we can write

\boxed{ a_{2} = -v \cdot \frac{ 1 }{ \sqrt{ ( 1 - v^{2}/c^{2} ) } } }

Using equation (8) we can write

b_{1}^{2} c^{2} = \frac{ 1 }{ (1 - v^{2}/c^{2} ) } - 1

b_{1}^{2} c^{2} = \frac{ 1 - ( 1 - v^{2}/c^{2} ) }{ (1 - v^{2}/c^{2} ) } = \frac{ v^{2}/c^{2} }{ (1 - v^{2}/c^{2} ) } = \frac{ v^{2} }{ c^{2} } \cdot \frac { 1 }{ (1 - v^{2}/c^{2} ) }

so b_{1}^{2} = \frac{ v^{2} }{ c^{4} } \cdot \frac{ 1 }{ ( 1 - v^{2}/c^{2} ) }

Taking the negative square root we can write

\boxed{ b_{1} = - \frac{ v }{ c^{2} } \cdot \frac{ 1 }{\sqrt{ (1 - v^{2}/c^{2} ) }} }

From equation (9) we can write

b_{2}^{2} c^{2} = c^{2} + v^{2} \cdot \frac{ 1 }{ ( 1 - v^{2}/c^{2} ) } = \frac{ c^{2}( 1 - v^{2}/c^{2} ) + v^{2} }{ ( 1 - v^{2}/c^{2} ) } = \frac{ c^{2} - v^{2} + v^{2} }{ (1 - v^{2}/c^{2} ) } = \frac{ c^{2} }{ ( 1 - v^{2}/c^{2} ) }

which leads to

b_{2}^{2} = \frac{ 1 }{ ( 1 - v^{2}/c^{2} ) }

and so

\boxed{ b_{2} = \frac{ 1 }{ \sqrt{ ( 1 - v^{2}/c^{2} ) } } }

which is the same as a_{1}.

If we define

\gamma = \frac{ 1 }{ \sqrt{ ( 1 - v^{2}/c^{2} ) } }

we can write

a_{1} = \gamma, \;\;\; a_{2} = -\gamma v, \;\;\; b_{1} = -\frac{ v }{ c^{2} } \cdot \gamma \rm{\;\;\ and \;\;\;} b_{2} = \gamma

Thus we can finally write our transformations as

\boxed {\begin{array}{lcl} x^{\prime} & = & \gamma (x - vt) \\ y^{\prime} & = & y \\ z^{\prime} & = & z \\ t^{\prime} & = & \gamma ( t - \frac{ v }{ c^{2} }x ) \end{array} }

These are known as the Lorentz transformations.

The Lorentz factor

The term \gamma is know as the Lorentz factor.

 

The Lorentz factor gamma plotted against speed as a fraction of the speed of light.

The Lorentz factor \gamma plotted against speed as a fraction of the speed of light.

 

As this plot shows, the Lorentz factor is essentially unity until the ratio v/c (the ratio of the speed to the speed of light) becomes about half of the speed of light, or about 1.5 \times 10^{8} m/s. Given that even our fastest space ships only travel at a tiny fraction of the speed of light, it is not surprising that we have no direct experience of the weird effects that a Lorentz factor deviating significantly from one produce. Of course we see these effects in particle accelerators and cosmic ray showers, but human beings are a long way from attaining speeds where the Lorentz factor will deviate from unity.

In a future blog I will discuss some of these weird effects. They include time passing more slowly and distances shrinking. Very very weird; but very very real, they are shown to happen every day in our particle accelerators.

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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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