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## Derivation of relativistic mass

I have taught special relativity for many years, but every time I teach it I present the result that mass changes as a function of velocity as a consequence of the modified version of Newton’s 2nd law.

As almost everyone knows, Newton’s 2nd law says that

$F=ma$

where $F$ is the force applied, $m$ is the mass, and $a$ is the acceleration felt by the body. In Newtonian mechanics, mass is invariant, but a consequence of special relativity is that nothing can travel faster than the speed of light $c$. This raises the conundrum of why can’t we keep applying a force to a body of mass $m$, causing it to continue accelerating and to ultimately increase its velocity to one greater than the speed of light?

The answer is that Newton’s 2nd law is incomplete. Einstein showed that mass is also a function of velocity, and so we should write

$m = \gamma m_{0} \text{ (1) }$

Where $\gamma = \frac{ 1 }{ \sqrt{ (1 - V^{2}/c^{2}) } }$ is the so-called Lorentz factor and$m_{0}$ is the rest mass (also known as the invariant mass or gravitational mass), the mass an object has when it is at rest relative to the observer. Hence we can argue that, as we approach the speed of light, the applied force goes into changing the mass of the body, rather than accelerating it, leading to a modified version of Newton’s 2nd law

$F = \gamma m_{0} a$

where both velocity and/or mass change as a force is applied. But, because of the fact that $\gamma \approx 1$ until $V \approx c/2$ (see Figure 1), very little increase in mass occurs until $V$ has reached appreciable values.

The variation of $\gamma$ (the Lorentz factor) as a function of the speed $V$. Until $V \approx c/2, \; \gamma$ is very close to unity

However, I have always found this an inadequate explanation of the relativistic mass, as it does not derive it but rather argues for its necessity. So, as I’m teaching special relativity again this year, I decided a few weeks ago to see if I could find a way of deriving it from a simple argument. After several weeks of hunting around I think I have found a derivation which is robust and easy to understand. But, in my searching I came across several “derivations” which were nothing more than circular arguments, and also some derivations which were simply incorrect.

## Two balls colliding

The best explanation that I have found to derive the relativistic mass is to use the scenario of two balls colliding. Although it would be possible, in theory to have the balls moving in any direction, we are going to make things a lot easier by having the balls moving in the y-direction, but with the two reference frames $S \text{ and } S^{\prime}$ moving relative to each other with a velocity $V$ in the x-direction. Also, the balls are going to have the same rest mass, $m_{0}$, as measured in their respective frames $S$ and $S^{\prime}$ (the rest mass of each ball can be measured by each observer in their respective reference frames when they are at rest in their respective frames).

Ball $A$ moves solely in the y-direction in reference frame $S$, and ball $B$ moves solely in the y$^{\prime}$-direction in reference frame $S^{\prime}$. Ball $A$ starts by moving in the positive y-direction in reference frame $S$ with a velocity $u_{0}$, and ball $B$ starts moving in the negative y$^{\prime}$-direction in reference frame $S^{\prime}$ with a velocity $-u_{0}$ in frame $S^{\prime}$.

Reference frame $S^{\prime}$ is moving relative to frame $S$ at a velocity $V$ in the positive x-direction. So, as seen in $S$, the motion of ball $B$ appears as shown in the left of Figure 2. That is, it appears in $S$ to move both in the negative y-direction and the positive x-direction, and so follows the path shown by the red arrow pointing downwards and to the right.

At some moment the two balls collide. After the collision, as seen in $S$, ball $A$ will move vertically downwards in the negative y-direction, with a velocity $-u_{0}$. Ball $B$ moves upwards (positive y-direction) and to the right (positive x-direction), as shown by the red arrow in the diagram on the left of Figure 1.

In reference frame $S^{\prime}$ the motions of balls $A$ and $B$ looks like the diagram on the right of Figure 1. In $S^{\prime}$, it is ball $B$ which moves vertically, and ball $A$ which moves in both the $x^{\prime}$ and $y^{\prime}$ directions.

Two balls colliding. Ball $A$ (in blue) moves solely in the y-direciton as seen in frame $S$, ball $B$ (in red) moves solely in the y-direction in frame $S^{\prime}$.

## The velocity of ball $B$ in $S$

To calculate the velocity of ball $B$ as seen in $S$, we have to use the Lorentz transformations for velocity. As we showed in this blog here, if we have an object moving with a velocity $u^{\prime}$ in $S^{\prime}$ which is moving relative to $S$ with a velocity $V$, then the velocity $u$ in frame $S$ is given by

$u = \frac{ u^{\prime} + V }{ \left( 1 + \frac{ u^{\prime}V }{ c^{2} } \right) } \text{ (2) }$

This equation is true when the velocity is in the x$^{\prime}$-direction, and the frames are moving relative to each other in the x-direction. So we are going to re-write Equ. (2) as

$u_{x} = \frac{ u^{\prime}_{x} + V }{ \left( 1 + \frac{ u^{\prime}_{x}V }{ c^{2} } \right) } \text{ (3) }$

However, if the velocity of an object is in the y$^{\prime}$-direction, rather than the x$^{\prime}$-direction, then we need a different expression. We can derive it from going back to our equations for the Lorentz transformations

The Lorentz transformations

This time we write

$dy = dy^{\prime}$

and

$dt = \gamma \left( dt^{\prime} + \frac{ dx^{\prime}V }{ c^{2} } \right)$

So

$\frac{ dy }{ dt } = \frac{ dy^{\prime} }{ \gamma \left( dt^{\prime} + \frac{ dx^{\prime}V }{ c^{2} } \right) }$

Dividing each term in the right-hand side by $dt^{\prime}$, we get

$\frac{ dy }{ dt } = \frac{ dy^{\prime}/dt^{\prime} }{ \gamma \left( dt^{\prime}/dt^{\prime} + \frac{ dx^{\prime}V }{ dt^{\prime}c^{2} } \right) }$

$u_{y} = \frac{ u^{\prime}_{y} }{ \gamma \left( 1 + \frac{ u^{\prime}_{x}V }{ c^{2} } \right) } \text{ (4) }$

Equations (3) and (4) allow us to work out the components of ball $B$’s velocities $u_{x}$ in the x-direction and $u_{y}$ in the y-direction in frame $S$.

$u(B)_{x} = \frac{ 0 + V }{ \left( 1 + \frac{ 0 \cdot V }{ c^{2} } \right) } = V \text{ (5) }$

$u(B)_{y} = \frac{ -u_{0} }{ \gamma \left( 1 + \frac{ 0 \cdot V }{ c^{2} } \right) } = \frac{ -u_{0} }{ \gamma } \text{ (6) }$

After the collision, the velocity of ball $A$ becomes $u(A) = -u_{0}$. What about ball $B$?

We can see that $u(B)_{x}$ will not change, and $u(B)_{y}$ after the collision will be $- \frac{ + u_{0} }{ \gamma }$.

## The momentum before and after the collision

We are now going to look at the momentum of balls $A$ and $B$ before and after the collision, as seen in frame $S$. We will start off by assuming that the mass is constant for both balls, that is that $m=m_{0}$ for both balls, despite the two reference frames moving relative to each other.

If we do this, we can write that the momentum in the x-direction before the collision is given by

$(p(A)_{x} + p(B)_{x})_{i} = 0 + m_{0}V = m_{0}V$

The momentum after the collision in the x-direction is given by

$(p(A)_{x} + p(B)_{x})_{f} = 0 + m_{0}V = m_{0}V$

So, momentum is conserved in the x-direction. But, what about in the y-direction? Before the collision, the momentum is given by

$(p(A)_{y} + p(B)_{y})_{i} = + m_{0}u_{0} + m_{0} \left( \frac{ -u_{0} }{ \gamma } \right) =m_{0}u_{0} - \frac{ m_{0}u_{0} }{ \gamma }$

After the collision, the momentum in the y-direction is given by

$(p(A)_{y} + p(B)_{y})_{f} = m_{0}(-u_{0}) + m_{0} \left( \frac{ +u_{0} }{ \gamma } \right) = -m_{0}u_{0} + \frac{ m_{0}u_{0} }{ \gamma }$.

If we assume that momentum is conserved, we can write

$m_{0}u_{0} - \frac{ m_{0}u_{0} }{ \gamma } = -m_{0}u_{0} + \frac{ m_{0}u_{0} }{ \gamma } \rightarrow 2m_{0}u_{0} = \frac{ 2m_{0}u_{0} }{ \gamma } \rightarrow \gamma = 1$

So, if we assume that the mass of both ball $A$ and ball $B$ in frame $S$ is $m_{0}$, the momentum in the y-direction is only conserved if $\gamma =1$. But, $\gamma$ is only equal to unity when the relative velocity $V$ between the two frames is zero; in other words when the two frames are not moving relative to each other! If $V \neq 0$ and mass is constant, momentum will not be conserved.

In physics, the conservation of momentum is considered a law, it is believed to always hold. In order for momentum to be conserved, we can qualitatively see that the mass of ball $B$ needs to be greater than the mass of ball $A$ as seen in frame $S$, as the speed of ball $B$ in the y-direction in frame $S, |u(B)_{y}| = u_{0} / \gamma < u_{0}$.

## Allowing the mass to change

We have just shown above that, if we assume both masses are invariant, momentum will only be conserved in the y-direction in the trivial case where the two frames are stationary relative to each other. So, let us now assume that, if $V \neq 0$, we have to allow the masses to change.

We will assume that mass is a function of speed. For ball $A$, the momentum in the x-direction is still zero, both before and after the collision. For ball $B$, we will now write the momentum in the x-direction, both before and after the collision, as

$p(B)_{x} = m(B) u(B)_{x} = m(B) V$

What about in the y-direction? For ball $A$, before the collision we can write

$p(A)_{y} = m(A) u(A)_{y} = m(A) u_{0}$

Where $m(A)$ is the mass of ball $A$ in frame $S$ which is affected by its velocity in frame $S$, which is $u_{0}$.

For ball $B$ as seen in frame $S$ we can write that the momentum in the y-direction before the collision is given by

$p(B)_{y} = m(B) u(B)_{y} = m(B) \cdot \left( \frac{ - u_{0} }{ \gamma } \right) = \frac{ -m(B)u_{0} }{ \gamma }$

Where $\gamma = 1/(1-V^{2}/c^{2})$, the Lorentz factor due to the relative velocity $V$ between $S$ and $S^{\prime}$.

After the collision, we can write the momentum for ball $A$ in the y-direction as being

$p(A)_{y} = m(A) u(A)_{y} = -m(A) u_{0}$

And, for ball $B$ we can write

$p(B)_{y} = m(B) u(B)_{y} = \frac{ +m(B)u_{0} }{ \gamma }$

Equating the momentum in the y-direction before and after the collision, we have

$m(A) u_{0} - \left( \frac{ m(B) u_{0} }{ \gamma } \right) = -m(A) u_{0} + \left( \frac{ m(B) u_{0} }{ \gamma } \right)$

$\rightarrow 2m(A) u_{0} = 2 \left( \frac{ m(B) u_{0} }{ \gamma } \right) \rightarrow m(A) =\frac{ m(B) }{ \gamma }$

For ball $A$, we will write

$m(A) = \gamma_{A} m_{0}$

where

$\gamma_{A} = \frac{ 1 }{ \sqrt( 1 - u^{2}(A)/c^{2} ) } = \frac{ 1 }{ \sqrt( 1 - u_{0}^{2}/c^{2} ) }$

(that is, $\gamma_{A}$ depends on the speed of ball $A$ in frame $S$, and that speed is $u_{0}$).

So, the momentum of ball $A$ in the y-direction is given by

$p(A)_{y} = m(A)u(A)_{y} \rightarrow \boxed {p(A)_{y} = \frac{ m_{0} u_{0} }{ \sqrt( 1 - u_{0}^{2}/c^{2} ) } \text{ (7) } }$

For ball $B$, we will write

$m(B) = \gamma_{B} m_{0}$

Where $\gamma_{B} = 1/\sqrt{ (1 - u^{2}(B)/c^{2} ) }$ depends on the speed $u(B)$ of ball $B$ as seen in frame $S$. (Note: the mass does not depend on just the y-component of ball $B$‘s speed (as is often incorrectly stated), it depends on its total speed).

To calculate the value of $u(B)$ we note that it is made up of the x-component $u(B)_{x}$ and the y-component $u(B)_{y}$. But, $u(B)_{x} = V$, and we showed above that $u(B)_{y} = -u_{0}/ \gamma$, where this $\gamma = 1/\sqrt{ (1 - V^{2}/c^{2}) }$.

Using Pythagoras to calculate $u(B)$, we have

$u(B)^{2} = V^{2} + u_{0}^{2}/\gamma^{2} = V^{2} + u_{0}^{2}(1 -V^{2}/c^{2})$

so

$u(B)^{2} = u_{0}^{2} + V^{2}( 1 - u_{0}^{2}/c^{2} )$

Using this value of $u(B)$ we can write

$\gamma_{B} = \frac{ 1 }{ \sqrt{ ( 1 - u(B)^{2}/c^{2} )} } = \frac{ 1 }{ \sqrt{ ( 1 - u_{0}^{2}/c^{2} - V^{2}/c^{2} + V^{2}u_{0}^{2}/c^{4} ) } }$

But, the terms $( 1 - u_{0}^{2}/c^{2} - V^{2}/c^{2} + V^{2}u_{0}^{2}/c^{4} )$ can be factorised as

$( 1 - u_{0}^{2}/c^{2} - V^{2}/c^{2} + V^{2}u_{0}^{2}/c^{4} ) = (1 - u_{0}^{2}/c^{2})(1 - V^{2}/c^{2})$

And so we can write

$\gamma_{B} = \frac{ 1 }{ \sqrt{ ( 1 - u_{0}^{2}/c^{2} ) } \sqrt{ (1 - V^{2}/c^{2}) } }$

But, $1/\sqrt{ (1 - V^{2}/c^{2}) } = \gamma$, so we can write

$\gamma_{B} = \gamma \cdot \frac{ 1 }{ \sqrt{ ( 1 - u_{0}^{2}/c^{2} ) } }$

This means that we can write the momentum for ball $B$ in the y-direction as

$p(B)_{y} = m(B) u(B)_{y} = \gamma_{B} m_{0} u(B)_{y}$

$p(B)_{y} = \gamma \cdot \frac{ 1 }{ \sqrt{ ( 1 - u_{0}^{2}/c^{2} ) } } \cdot m_{0} \cdot \frac{ u_{0} }{ \gamma }$

$\boxed{ p(B)_{y} = \frac{ m_{0}u_{0} }{ \sqrt{ ( 1 - u_{0}^{2}/c^{2} ) } } \text{ (8) } }$

Comparing this to Equ. (7), the equation for $p(A)_{y}$, we can see that they are equal, as required.

So, we have proved that, to conserve momentum, we need mass to be a function of speed, and specifically that

$\boxed{ m = \frac{ m_{0} }{ \sqrt{ (1 - u^{2}/c^{2}) } } }$

Where $u$ is the speed of the ball in a particular direction in frame $S$.

## How to add velocities in special relativity

As I mentioned in this blogpost, in special relativity any observer will measure the speed of light in a vacuum to be $c$, irrespective of whether the observer is moving towards or away from the source of light. We can think of the speed of light as a cosmic speed limit, nothing can travel faster than it.

But, let us suppose that we have two reference frames $S$ and $S^{\prime}$ moving relative to each other with a speed of $v=0.9c$, 90% of the speed of light. Surely, if someone in frame $S^{\prime}$ fires a high-speed bullet at a speed of $u^{\prime}= 0.6c$, an observer in frame $S$ will think that the bullet is moving away from him at a speed of $u = v + u^{\prime} = 0.9c + 0.6c = 1.5c$, which seemingly violates the comic speed limit.

What have we done wrong?

We cannot simply add velocities, as we would do in Newtonian mechanics. In special relativity we have to use the Lorentz transformations to add velocities. How do we do this? Let us remind ourselves that the Lorentz transformations can be written as

The Lorentz transformations to go either from reference frame $S \text{ to } S^{\prime}$, or to go from $S^{\prime} \text{ to } S$.

## Calculating a velocity in two different reference frames

To calculate the velocity $u$ of some object moving with a velocity $u^{\prime}$ in reference frame $S^{\prime}$ we need to use these Lorentz transformations.

We start off by writing

$x = \gamma \left( x^{\prime} + vt^{\prime} \right) \text{ (1) }$

and

$t = \gamma \left( t^{\prime} + \frac{x^{\prime}v}{c^{2} } \right) \text{ (2) }$

We will now take the derivative of each term, so we have

$dx = \gamma \left( dx^{\prime} + vdt^{\prime} \right)$

and

$dt = \gamma \left( dt^{\prime} + \frac{dx^{\prime}v}{c^{2} } \right)$

We can now write $dx/dt = u$ (the velocity of the object as seen in frame $S$) as

$\frac{dx}{dt} = \frac{ \gamma \left( dx^{\prime} + vdt^{\prime} \right) }{ \gamma \left( dt^{\prime} + \frac{dx^{\prime}v}{c^{2} } \right) }$

The $\gamma$ terms cancel, and dividing each term on the right hand side by $dt^{\prime}$ gives

$\frac{dx}{dt} = \frac{ \left( dx^{\prime}/dt^{\prime} + vdt^{\prime}/dt^{\prime} \right) }{ \left( dt^{\prime}/dt^{\prime} + \frac{dx^{\prime}v}{c^{2} dt^{\prime} } \right) } = \frac{ \left( u^{\prime} + v \right) }{ \left( 1 + \frac{ u^{\prime} v}{c^{2} } \right) }$

$\boxed{ u = \frac{ \left( u^{\prime} + v \right) }{ \left( 1 + \frac{ u^{\prime} v}{c^{2} } \right) } }$

where $u^{\prime}$ was the velocity of the object in reference frame $S^{\prime}$.

Going back to our example of $v = 0.9c$ and $u^{\prime} = 0.6c$, we can see that the velocity $u$ as measured by an observer in reference frame $S$ will be

$u = \frac{ 0.6c + 0.9c }{ \left( 1 + \frac{ (0.6c \times 0.9c) }{ c^{2} } \right) } = \frac{ 1.5c }{ 1 + 0.54 } = \frac{ 1.5c }{ 1.54} = \boxed {0.974c}$, not $1.5c$ as we naively calculated.

## The constancy of the speed of light

What happens if a person in reference frame $S^{\prime}$ shines a light in the same direction as $S^{\prime}$ is moving away from $S$? In this case, $u^{\prime}=1.0c$. Putting this into our equation for $u$ we get

$u = \frac{ 0.6c + 1.0c }{ \left( 1 + \frac{ (0.6c \times 1.0c) }{ c^{2} } \right) } = \frac{ 1.6c }{ 1 + 0.6 } = \frac{ 1.6c }{ 1.6} = 1.0c$

So they both agree that the light is moving away from them with the same speed $c$!

## Should Justin Gatlin be allowed to compete?

More pertinent today than ever…..

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?

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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## Can the Lions roar in New Zealand?

Tomorrow morning (Disunited Kingdom and Namibian time) the British & Irish Lions will play the 1st Test of their 2017 tour against New Zealand’s All Blacks. It is a much anticipated match, one of the most important in which most of the players will take part.

My first rugby memory is of the 1971 Lions tour of NZ, the only time that the Lions have won a Test series there. Of course, as a 7-year old I did not realise that the 1971 tour was, and would remain, so historic.

I didn’t say that I was witnessing history, as live TV of sporting events occurring on the other side of the world did not exist in 1971. At least, not for rugby. So we listened live on the radio, at 4am (as matches were played mid-afternoon local time). Later in the day, highlights would be shown on TV, which we also watched even though we knew the score.

The “we” were myself, my two sisters and my parents. My father gave us no choice in whether we were woken up to listen to the Test matches in 1971; refusal was not an option.

The 1971 tour remains burned in the minds of so many rugby fans, and not just supportes of the Lions but those of the All Blacks too. Many Kiwis of my age will tell you that their rugby heroes were Barry John, Gareth Edwards, Gerald Davies or J.P.R. Williams, not the men who represented New Zealand at that time.

This 2017 tour is not only a chance to get that second series win, It is also a chance for Kiwi Warren Gatland to put one over on his native country and put himself at the head of the list to succeed Steve Hansen as the next All Blacks’ coach. Here is the team chosen for the 1st Test.

To many people’s surprise, Leigh Halfpenny and George North have not been included. Gatland and his coaching team have, instead, gone with a far more attacking back three. The other major talking point is Allan Wyn Jones’ inclusion ahead of Maro Itoje, but I’m not surprised by this. Although Itoje has been in great form, Alun Wyn is one of the most experienced and best locks in World rugby. Tour captain Sam Warburton had pretty much declared himself out of contention to start the 1st Test, feeling that he’s not yet fit enough.

If the Lions can win the 1st Test then it will be a huge result. Not only will it make winning the 3-Test series a real possibility, but it will also be the first time that the All Blacks have lost at Eden Park since 1994! A loss by the All Blacks would send shockwaves through New Zealand rugby. The Lions need no greater motivation to play the game of their lives.

## Bob Dylan : 30 greatest songs (Daily Telegraph) – part 5 – 10 to 6

Continuing my countdown of the 30 greatest Bob Dylan songs according to the Daily Telegraph, today I am covering the songs from numbers 10 to 6. These are

• 10 – Knockin’ On Heaven’s Door
• 9 – Ballad Of A Thin Man
• 8 – Hurricane
• 7 – Visions Of Johanna
• 6 – Like A Rolling Stone

Bob Dylan granted his first interview since being awarded the 2016 Nobel prize in literature to Edna Gundersen.

### 10. Knockin’ On Heaven’s Door (1973)

This song has become very well known through cover versions, in particular the versions by Bob Marley and Guns ‘n’ Roses. As with most cover versions of Dylan songs, I have to say that I prefer the original (but I also know that I’m biased!). There is a simplicity and starkness to Dylan’s original, which is lost in the two more famous cover versions.  After saying that, I do think both cover versions are great, and the Guns ‘n’ Roses version is one of the few songs done by them that I like.

From the soundtrack of a violent Sam Peckinpah western, in which Dylan once again demonstrated that acting is not one of his many talents, comes this elegiac classic. It rides on a simple, repetitive chord progression and has a ridiculously swift fade out but conveys such a spirit of bittersweet farewell to life it has become one of rock’s most universal anthems.

### 9. Ballad Of A Thin Man (1965)

This song is believed to be a cutting criticism of an out-of-touch newspaper reporter, and was written during a period when Dylan was showing a different side to his song writing. A side in which he was laying bare his frustrations with the people who didn’t understand him, or the societal changes which he was spearheading. When he sang this song in concert during his infamous world tour of 1965-66, the anger in his voice was clear to all who listened (rather than those who were booing him for “going electric”).

It is comical to consider that Sixties Dylan is so associated with the peace and love ethos of the hippies. Over an ungainly, almost lumpen piano motif, Ballad Of A Thin Man heaps surreal scorn on some self-regarding representative of the straight world baffled by the inscrutable counter-culture. Dylan’s vocal drips contempt. “Something is happening and you don’t know what it is, do you, Mr Jones?” He sounds like the original punk.

### 8. Hurricane (1976)

Is there a songwriter who can weave a story as masterfully as Dylan? The story in this song, however, is shockingly true. It tells of the incorrect conviction and imprisonment of Ruben Carter, a boxer who was also known as the Hurricane. The song is the opening track on Dylan’s 1976 album Desire, which is one of my favourite Dylan albums. A movie was later made of Carter’s life, with this as the opening song.

“Pistol shots ring out in the barroom night…” Hurricane marked a thrilling late flourish from Dylan the protest singer, moved to write by the flagrant framing of champion black boxer Ruben Carter (finally exonerated in 1985). The dramatic temperature of this forensically bitter narrative (composed with Jaques Levy) is matched by wild violin flourishes from beautiful novice Scarlet Rivera, who Dylan picked up walking in the street on the way to the recording session.

### 7. Visions of Johanna (1966)

This song is simply mesmerising. The opening lines “Ain’t it just like the night to play tricks / When you’re trying to be so quiet” are masterful. Although I like the studio version which appears on Blonde on Blonde a lot; when I heard the acoustic version that he did in concert in May 1966 in Manchester (the famous “Royal Albert Hall” concert, with the Judas heckle), I was simply blown away by the haunting power of that live version. Bear in mind that, when he performed this in May 1966, the audience didn’t know the song at all as Blonde on Blonde had not yet been released.

“Inside the museums, infinity goes up on trial.” Dylan at his most expressive and elusive, slipping in and out of the cracks of his own lyrics as he holds contrasting romantic muses in the balance. “I do know what my songs are about,” he insisted to an interviewer from Playboy magazine. “Some are about four minutes, some are about five minutes, and some, believe it or not, are about eleven or twelve.

### 6. Like A Rolling Stone (1965)

In Rolling Stone Magazine’s list of Dylan’s greatest songs, as well as their list of the greatest songs of all time, “Like A Rolling Stone” is at number 1. The Telegraph puts it at number 6, so you will have to wait and see which songs they put above it. When released in 1965, this song was the longest song ever released as a single. It was also Dylan’s biggest chart success. He finished the set of each concert in his 1965-66 world tour with it. Sometimes, the incessant booing which accompanied the second half of the show (the “electric” half) would cease during this song, as it was such a big hit in the USA and Europe.

“That snare shot sounded like somebody’d kicked open the door to your mind” is how Bruce Springsteen recalled first hearing this at 15 years old. This thunderous six-minute rock epic marks the moment when the young protest singer emerged as something popular music had never witnessed before. The vocal is as fierce and relentless as the flowing, spitting lyric, a tale of a fallen society princess adjusting to a disorientating new reality. “How does it feeeel?” Dylan demands. Many of us are still wondering about that.

## Knockin’ on Heaven’s Door

The song which I have chosen to share today is “Knockin’ on Heaven’s Door”, as it is the only one which is on Dylan’s official Vevo channel. This version is from his MTV Unplugged concert, which he did in the mid-1990s. Enjoy!

## Wales v Ireland preview

Tonight (Friday 10th March), Wales take on Ireland in Cardiff in the 2017 6 Nations clash. It is the first match of the penultimate weekend of this year’s 6 Nations, and a Friday evening clash in Cardiff usually has a special atmosphere. Wales have lost two matches from three, with their heavy defeat at Murrayfield two weekends ago still being picked over by the rugby media and pundits in Wales. This means that we are out of contention for this year’s 6 Nations title, but there is more than pride at stake.

The 2019 World Cup draw will be made in the next few months, and if Wales do not win at least one of their remaining two matches in the 6 Nations they risk dropping out of the top 8 in the World rankings. This means that they will be drawn in the same group as two higher ranked teams, with one team from the top 4 and the second team from 5-8. This was the situation we faced in the 2015 World Cup, being drawn in the same group as Australia and England.

This has left Welsh interim-coach Rob Howley with a difficult decision to make. There have been calls for him to try starting players like Sam Davies at outside half, with the 6 Nations title now out of reach. But, we cannot afford to lose either of our two remaining matches because of the World Cup draw. So, despite most people in Wales calling for some changes to the starting 15, Howley has decided to keep exactly the same 23 as he did against Scotland, with the same starting 15 too.

Wales take on Ireland this evening in Cardiff. Wales will be looking to salvage a very poor 2017 6 Nations campaign.

Ireland, on the other hand, are still in with a chance of the 6 Nations title. Win in Cardiff tonight and it will come down to a probable title-decider against England at home in Dublin on Saturday the 18th. Based on their performances in their 3 matches so far, I think Ireland will be too good for Wales. Of course I want to see a Wales victory, but unfortunately I think Ireland are going to win. Whoever wins, it should be a very exciting match.

## 2017 6 Nations – 3rd weekend review

The 3rd weekend of the 2017 6 Nations saw home victories in all three matches. Scotland beat Wales at Murrayfield, Ireland beat France in Dublin, and England beat Italy at Twickenham.

## Scotland v Wales

Scotland beat Wales for the first time since 2007, with a well-deserved win that saw them score 20 unanswered points in the 2nd half. Wales went into half-time with a slim 13-9 lead. It could have been more, with Wales missing a chance to go 16-6 up with only a few minutes left of the first half. Instead, Scotland scored a penalty just before half-time and instead of 16-6 it was 13-9.

If ever there was a game of two halves this was it. The second half saw Scotland score 20 unanswered points, including two well deserved tries. It is true that Wales did have their chances to score points in the 2nd half, but Scotland were by far the better team after the break and fully deserved their victory over a largely ineffective Welsh team.

Scotland beat Wales 29-13 in a well-deserved victory, their first win over Wales since 2007.

Whereas the Scottish team took their chances, Wales were unable to come away with points when they were in the Scottish 22. It was a poor performance by Wales, but also a very good one by Scotland. After being uncompetitive for most of the last 10 years, it is good to see Scotland back mixing it with the other countries in the 6 Nations.

## Ireland v France

This was a cracker of a match. For a neutral, it had everything. Ireland went into half-time with a narrow 7-6 lead, and it was not clear well into the second half who was going to win this keenly contested match. In the end the better play of Irish half-backs Connor Murray and Johnny Sexton proved decisive, Ireland emerging victorious 19-9. With first Ireland, and then France, being the last two sides that Wales will face, the performance of both teams in this match does not bode well for Wales’ chances of winning either match, in my opinion.

## England v Italy

I have not seen this match, but am looking forward to seeing the highlights, mainly for the tactics which Italy adopted. By not committing any men to the breakdown, they avoided any rucks forming and thus the off-side line which normally exists did not apply. This allowed them to have men standing between the English backs, completely disrupting the game that England hoped to play. In fact, so effective was this tactic that Italy were 10-5 ahead at half-time. It is, apparently, only in the second half that England got to grips with this novel tactic, and eventually ran away to win 36-15.

## The 4th weekend

The 6 Nations now has another brief hiatus, before the next round of matches in just under two weeks. Ireland come to Cardiff to play Wales on Friday evening (10th March), then on the Saturday Italy take on France before the big game of the weekend, England v Scotland. Not only is this for the Calcutta Cup, but both countries are going for the Triple Crown. It will  be the first time in probably some 20 years that Scotland will go to Twickenham with even the slightest hope of winning.

Based on our performance against Scotland, I cannot see Wales beating Ireland, which leaves us with the real possibility of finishing the 2017 6 Nations with only one win. Quite where this will leave the Welsh team and management is a good question,  but it is clear from our performances this season that Wales have not developed the creativity in attack which is needed to win matches. Scotland showed us how it should be done; all we could do was look on and marvel at how much they have developed and how much we have regressed.