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the discovery of the electron

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5. Simple Twist Of Fate (1975)

Dylan’s saddest song. As he sings of the last night of a couple together with weary despair, Dylan’s narrative unfolds in the third person, except for one devastating giveaway “I remember well”. Even the wheezy harmonica solo sounds emotionally shattered.

4. Jokerman (1983)

Many of Dylan’s most devoted fans were alienated by the preachiness of Dylan’s born again Christian phase. On Jokerman he released himself back into a beautiful ambiguity that more perfectly distils the mysteries of faith. The Jokerman is Jesus, “born with a snake in both of your fists while a hurricane was blowing.” With legendary Jamaican rhythm section Sly Dunbar and Robbie Shakespeare pulsing beneath Mark Knopfler’s silvery guitar, the track has a slipperiness that mirrors its audacious lyrical twists and turns.

3. A Hard Rain’s A Gonna Fall (1963)

Adapting the melody and refrain of traditional English folk song Lord Randall, Dylan lets loose the full force of his poetic imagination like an apocalyptic storm. This first recording is sparse, just strummed acoustic guitar and that barbed wire voice, but the cascade of imagery holds listeners in hypnotic thrall.

2. Blowin’ In the Wind (1963)

This song was released in 1962 by Dylan on his Freewheelin’ Bob Dylan album.

The song that really announced Dylan, a troubadour for a generation, old beyond his years. He was just 21 when he wrote it but cautiously held it back from his debut album of folk covers. It sounds like a song that has been blowing around for 1000 years and has a quality of simple, universal, mysterious truth that will keep it relevant for a thousand more.

1. Tangled Up In Blue (1975)

The most dazzling lyric ever written, an abstract narrative of relationships told in an amorphous blend of first and third person, rolling past, present and future together, spilling out in tripping cadences and audacious internal rhymes, ripe with sharply turned images and observations and filled with a painfully desperate longing. “I wanted to defy time” according to Dylan. “When you look at a painting, you can see any part of it altogether. I wanted that song to be like a painting.”

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Expanding spheres of light

In this blogpost here, I derived the Lorentz transformations from first principles. The derivation uses a simple thought experiment; two reference frames S \text{ and } S^{\prime} are moving relative to each other with a speed v. The origins of S \text{ and } S^{\prime} coincide spatially at time t=t^{\prime}=0. At this moment, a flash of light is created at their origins, and expands as a sphere.


The radius of this sphere of light will be r \text{ and } r^{\prime} in S \text{ and } S^{\prime} respectively. So, we can write

x^{2} + y^{2} + z^{2} = r^{2} \text{ in } S \text{ (1) }
and
x^{\prime}{^2} + y^{\prime}{^2} +z^{\prime}{^2} = r^{\prime}{^2} \text{ in } S^{\prime} \text{ (2) }

But, in reference frame S, the distance r that light travels in time t is related to the speed of light c, as c = r/t, so for S we can write that

r = ct

We can do the same thing for reference frame S^{\prime}. The distance r^{\prime} that light travels in time t^{\prime} in S^{\prime} is related to the speed of light c^{\prime} in S^{\prime}, c^{\prime} = r^{\prime} / t^{\prime}, so for S^{\prime} we can write

r^{\prime} = c^{\prime} t^{\prime}.

However, crucially, Einstein said that the speed of light c is the same for all inertial observers. This is the most important principle which underpins special relativity. It means that c^{\prime} = c, and so we can write

r^{\prime} = c t^{\prime}

which allows us to write Equations (1) and (2) as

x^{2} + y^{2} + z^{2} -c^{2}t^{2} = 0 \text{ in } S \text{ (3) }

and

x^{\prime}{^2} + y^{\prime}{^2} + z^{\prime}{^2} - c^{2}t^{\prime}{^2} = 0 \text{ in } S^{\prime} \text{ (4) }

Using the Galilean Transformations, which can be written as


we can substitute xfor x^{\prime} (=x - vt), y^{\prime}=y, z^{\prime}=z and t^{\prime} =t in Equ. (4) to give us

(x-vt)^{2} + y^{2} + z^{2} - c^{2}t^{2} = x^{2} - 2vxt +v^{2}t^{2} + y^{2} + z^{2} - c^{2}t^{2}

which is the same  as Equ. (3), except for the two extra terms -2vxt \text{ and } v^{2}t^{2}. This shows that Equations (3) and (4) are not equal under a Galilean transformation, even though the two equations should be equal (as both are equal to zero).

Let us now show that Equations (3) and (4) are equal using the Lorentz transformations. These can be written as


where \gamma is the Lorentz factor, and is defined as

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

We will substitute these expressions for x^{\prime},y^{\prime},z^{\prime} and t^{\prime} into Equ. (4). When we do this, we have

\left( \gamma (x-vt) \right)^{2} + y^{2} + z^{2} - c^{2} \left( \gamma (t - vx/c^{2}) \right)^{2}

Multiplying this out, and dropping the y^{2} and z^{2} terms, we get

\gamma^{2}(x^{2} -2vxt +v^{2}t^{2}) -c^{2}\gamma^{2}\left(t^{2} - \frac{2vxt}{c^{2}} + \frac{v^{2}x^{2}}{c^{4}}\right)

\gamma^{2}x^{2} -2\gamma^{2}vxt + \gamma^{2}v^{2}t^{2} - c^{2}\gamma^{2}t^{2} +2\gamma^{2}vxt - (\gamma^{2}v^{2}x^{2})/c^{2}

The two terms 2\gamma^{2} vxt disappear and, gathering terms in x^{2} and t^{2} together, we can write

\gamma^{2}x^{2} - \left( \frac{ (\gamma^{2}v^{2}) }{ c^{2} } \right) x^{2} + \gamma^{2}v^{2}t^{2} - \gamma^{2}c^{2}t^{2}

\gamma^{2}x^{2} (1 - v^{2}/c^{2}) + \gamma^{2}t^{2}(v^{2}-c^{2})

Remembering that \gamma^{2} = \frac{ 1 }{ ( 1 - v^{2}/c^{2} ) }, and changing the sign of the t^{2} term, we can write

\frac{ 1 }{ (1 - v^{2}/c^{2} ) } \cdot (1 - v^{2}/c^{2}) \cdot x^{2} - \frac{ 1 }{ (1 - v^{2}/c^{2} ) } \cdot (c^{2} - v^{2}) \cdot t^{2}

which is

x^{2} - \frac{ 1 }{ (1 - v^{2}/c^{2} ) } \cdot c^{2}(1 - v^{2}/c^{2}) \cdot t^{2}

which is

x^{2} - c^{2}t^{2}, exactly as in Equ. (3).

We have therefore shown that Equations (3) and (4) are equal if we use the Lorentz transformations.

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Today (January 30th) marks the 50th anniversary of the last time The Beatles played live together, in the infamous “rooftop” concert in 1969. Although they would go on to make one more studio album, Abbey Road in the summer of 1969; due to contractual and legal wranglings the rooftop concert, which was meant to be the conclusion to the movie they were shooting, would not come out until 1970 in the movie Let it Be.

It is also true to say that some of the songs on Abbey Road were performed “live” in the studio with very little overdubbing (as opposed to separate instrument parts being recorded separately as was done on e.g. Sgt. Pepper). But, the rooftop concert was the last time the greatest band in history were seen playing together, and has gone down in infamy. It has been copied by many, including the Irish band U2 who did a similar thing to record the video for their single “Where the Streets Have no Name” in 1987 in Los Angeles.

The Beatles were trying to think of a way to finish the movie that they had been shooting throughout January of 1969. They had discussed doing a live performance in all kinds of places; including on a boat, in the Roundhouse in London, and even in an amphitheatre in Greece. Finally, a few days before January 30th 1969, the idea of playing on the roof of their central-London offices was discussed. Whilst Paul and Ringo were in favour of this idea, and John was neutral, George was against it.

The decision to go ahead with playing on the roof was not made until the actual day. They took their equipment up onto the roof of their London offices at 3, Saville Row, and just start playing. No announcement was made, only The Beatles and their inner circle knew about the impromptu concert.

The concert consisted of the following songs :

  1. “Get Back” (take one)
  2. “Get Back” (take two)
  3. “Don’t Let Me Down” (take one)
  4. “I’ve Got a Feeling” (take one)
  5. “One After 909”
  6. “Dig a Pony”
  7. “I’ve Got a Feeling” (take two)
  8. “Don’t Let Me Down” (take two)
  9. “Get Back” (take three)

People in the streets below initially had no idea what the music (“noise”) coming from the top of the building was, but of course younger people knew the building was the Beatles’ offices. However, they would not have recognised any of the songs, as these were not to come out for many more months. After the third song “Don’t Let Me Down”, the Police were called and came to shut the concert down. The band managed nine songs (five different songs, with three takes of “Get Back”, two takes of “Don’t Let Me Down”, and two takes of “I’ve Got a Feeling”) before the Police stopped them. Ringo Starr later said that he wanted to be dragged away from his drums by the Police, but no such dramatic ending happened.

At the end of the set John said

I’d like to thank you on behalf of the group and ourselves, and I hope we’ve passed the audition.

You can read more about the rooftop concert here.

Here is a YouTube video of “Get Back” (which may get taken down at any moment)

 

 

and here is a video on the Daily Motion website of the whole rooftop concert (again, it may get taken down at any moment).

 

 

Enjoy watching the greatest band ever perform live for the very last time!

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In late October 2017, astronomers announced the first ever discovery of an asteroid (or comet?) coming into our Solar System from another stellar system. The object was first spotted on 19 October by the University of Hawaii’s Pan-STARRS telescope, during its nightly search for near-earth objects. Based on its extreme orbit and its rapid speed, it was soon determined that the object has come into our Solar System from somewhere else, and this makes it the first ever asteroid/comet with an extra-solar origin to have been discovered. Originally given the designation A/2017 U1, the International Astronomical Union (IAU) have now renamed it 1I/2017 U1, with the I standing for “interstellar”. 

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The object, given the designation A/2017 U1, was deemed to be extra-solar in origin from an analysis of its motion.

In addition to its strange trajectory, observations suggest that the object also has quite an unusual shape. It is very elongated, being ten times longer than it is wide. It is thought to be at least 400 metres long but only about 40 metres wide. This was determined by the rapid and dramatic changes in its brightness, which can only be explained by an elongated object tumbling rapidly.

The object has also been given the name Oumuamua (pronounced oh MOO-uh MOO-uh), although this is not its official name (yet).  This means “a messenger from afar arriving first” in Hawaiian. In other respects, it seems to be very much like asteroids found in our own Solar System, and is the confirmation of what astronomers have long suspected, that small objects which formed around other stars can end up wandering through space, not attached to any particular stellar system.

To read more about this fascinating object, follow this link.

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Derivation of E=mc2

There are quite a few ways to derive Einstein’s famous equation E=mc^{2}. I am going to show you what I consider to be the simplest way.  Feel free to comment if you think you know of an easier way.

We will start off with the relationship between energy, force and distance. We can write

dE = F dx \text{ (1) }

Where dE is the change in energy, F is the force and dx is the distance through which the object moves under that force.  But, force can also be written as the rate of change of momentum,

F = \frac{dp}{dt}

Allowing us to re-write Equation (1) as

dE = \frac{dp}{dt}dx \rightarrow dE = dp \frac{dx}{dt} = vdp \text{ (2) }

Remember that momentum p is defined as

p =mv

In classical physics, mass is constant. But this is not the case in Special Relativity, where mass is a function of velocity (so-called relativistic mass).

m = \frac{ m_{0} }{ \sqrt{ ( 1 - v^{2}/c^{2} ) } } \text{ (3) }

where m_{0} is defined as the rest mass (the mass of an object as measured in a reference frame where it is stationary).

Assuming that both m \text{ and } v can change, we can therefore write

dp =mdv + vdm

This allows us to write Equ. (2) as

dE = vdp = v(mdv + vdm) = mvdv + v^{2}dm \text{ (4) }

Differentiating Equ. (3) with respect to velocity we get

\frac{dm}{dv} = \frac{d}{dv} \left( \frac{ m_{0} }{ \sqrt{ (1 - v^{2}/c^{2}) } } \right) = m_{0} \frac{d}{dv} (1 - v^{2}/c^{2})^{-1/2}

Using the chain rule to differentiate this, we have

\frac{dm}{dv} = m_{0} \cdot - \frac{1}{2} (1 - v^{2}/c^{2})^{-3/2} \cdot (-2v/c^{2}) = m_{0}  (v/c^{2}) \cdot (1 - v^{2}/c^{2})^{-3/2} \text{ (5) }

But, we can write

(1 - v^{2}/c^{2})^{-3/2} as (1-v^{2}/c^{2})^{-1/2} \cdot (1-v^{2}/c^{2})^{-1}

This allows us to write Equ. (5) as

\frac{dm}{dv} = m_{0}  (v/c^{2}) \cdot (1 - v^{2}/c^{2})^{-1} \cdot (1 - v^{2}/c^{2})^{-1/2}

From the definition of the relativistic mass in Equ. (3), we can rewrite this as

\frac{dm}{dv} = \frac{ m v }{ c^{2} }(1-v^{2}/c^{2})^{-1}

Which is

\frac{dm}{dv} = \frac{ m v }{ c^{2} } \left( \frac{c^{2}}{c^{2}} - \frac{ v^{2}}{c^{2} } \right)^{-1} = \frac{ m v }{ c^{2} } \left( \frac{c^{2}-v^{2}}{c^{2}} \right)^{-1}  = \frac{ m v }{ c^{2} } \left( \frac{c^{2}}{c^{2}-v^{2}}   \right)

\frac{dm}{dv} = \frac{ m v }{ (c^{2}-v^{2}) } \text{ (6) }

So we can write

c^{2}dm - v^{2}dm = mvdv

Substituting this expression for mvdv into Equ. (4) we have

dE = vdp = vd(mv) = mvdv + v^{2}dm = c^{2}dm - v^{2}dm + v^{2}dm

So

dE = c^{2} dm

Integrating this we get

\int_{E_{0}}^{E} dE = c^{2} \int_{m_{0}}^{m} dm

So

E - E_{0} = c^{2} ( m - m_{0} ) = mc^{2} - m_{0}c^{2}

E - E_{0} = mc^{2} - m_{0}c^{2}

This tells us that an object has rest mass energy E_{0} = m_{0}c^{2} and that its total energy is given by

\boxed{ E = mc^{2} }

where m is the relativistic mass.

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More pertinent today than ever…..

thecuriousastronomer

It was announced a few days ago that the American sprinter Justin Gatlin is on the shortlist for the IAAF (International Association of Athletics Federations) “Athlete of the Year” award for 2014. This is largely due to his having set the fastest times over both 100m and 200m this year; faster than Usain Bolt, faster than Yohan Blake, faster than anyone. In fact, he has set 6 of the 7 fastest times over 100m in 2014! Also, he has run faster over both 100m and 200m than anyone one else in their 30s (he is 32). Ever. But, should Gatlin be considered by the IAAF for such a prestigious award? Should he be even allowed to compete at all?

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For those of you not familiar with Gatlin’s athletics career, he has twice been banned for failing drugs tests. In 2001 he failed a doping test, testing positive for amphetamines. He…

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