The radius of this sphere of light will be in respectively. So, we can write

and

But, in reference frame , the distance that light travels in time is related to the speed of light , as , so for we can write that

We can do the same thing for reference frame . The distance that light travels in time in is related to the speed of light in , , so for we can write

.

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

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

and

Using the Galilean Transformations, which can be written as

we can substitute for , , and in Equ. (4) to give us

which is the same as Equ. (3), except for the two extra terms . 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 is the Lorentz factor, and is defined as

We will substitute these expressions for and into Equ. (4). When we do this, we have

Multiplying this out, and dropping the and terms, we get

The two terms disappear and, gathering terms in and together, we can write

Remembering that , and changing the sign of the term, we can write

which is

which is

, exactly as in Equ. (3).

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

In 1928 Dirac wrote a paper in which he published what we now call the Dirac Equation.

This is a relativistic form of Schrödinger’s wave equation for an electron. The wave equation was published by Erwin Schrödinger two years earlier in 1926, and describes how the quantum state of a physical system changes with time.

The various terms in this equation need some explaining. Starting with the terms to the left of the equality, and going from left to right, we have is the imaginary number, remember . The next term is just Planck’s constant divided by two times pi, i.e. . The next term is the partial derivative with respect to time of the wave function .

Now, moving to the right hand side of the equality, we have

which is the mass of the particle, is its potential energy, is the Laplacian. The Laplacian, is simply the divergence of the gradient of the wave function, .

In plain language, what the Schrödinger equation means “total energy equals kinetic energy plus potential energy”, but the terms take unfamiliar forms for reasons explained below.

]]>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 :

*“Get Back”*(take one)*“Get Back”*(take two)*“Don’t Let Me Down”*(take one)*“I’ve Got a Feeling”*(take one)*“One After 909”**“Dig a Pony”**“I’ve Got a Feeling”*(take two)*“Don’t Let Me Down”*(take two)*“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!

]]>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.

]]>On the evening of Saturday 17th November I booked a ticket to go up the Empire State Building to take night-time photos. The cost of a ticket to the 80th floor is US$32, and I’d say that it’s good value. But, what is not is the extra $20 to go up to the 102nd floor. I had forgotten from my last time that this is not worth the extra money. Hopefully I’ll remember next time.

I was on the observation deck of the Empire State (86th floor) for about 2 hours taking photos and videos. I’ll post some of them over the next few weeks. Then, at about 11:30pm I walked to Time Square. I stopped to get a coffee and warm up a bit so got to Time Square at about 12:30am. The place was heaving, hundreds of people were milling around and many of the shops were open.

At about 1:30am I caught the subway to Brooklyn as I wanted to get a photo of Manhattan with the Brooklyn Bridge in the foreground. I took my photos from the Brooklyn Bridge Park, then *walked* back to Manhattan over the Brooklyn Bridge.

By this time it was lashing it down with rain and I was soaked. The rain had seeped through my winter coat and my sweater and trousers were pretty much wet through. But, as I walked over the bridge I caught sight of the Statue of Liberty illuminated (it was about 3am by this time). So I decided to walk to Battery Park to take photos of her at night.

When I got to Battery Park and set up my camera I discovered that my camera battery was dead from all the long exposure photos I’d been taking. So I got my spare battery out of my backpack, only to find that it too was dead. So, I didn’t get any photos of Lady Liberty at night. It was now 4am.

I then walked back to my hotel which was in the Little Italy part of Manhattan. The rain was still lashing it down, and by now my phone was getting damp leading to Google maps misbehaving. The app kept on going haywire every minute or two, so I couldn’t use it to guide me back from Battery Park to Little Italy. Instead I just tried to figure it out, and it took me two hours!

I collapsed into my bed at 6am, having spent nearly 12 hours wandering around nighttime Manhattan taking photos.

When I was walking across the Brooklyn Bridge at about 3am this great Simon & Garfunkel song kept playing in my head.

This song appears on Simon & Garfunkel’s last album *Bridge Over Troubled Water*. Written, of course, by Paul Simon, the “Tom” in the lyrics refers to Art Garfunkel. When they were teenagers in Queens they released a single and briefly called themselves”Tom & Jerry”.

By 1970 Simon and Garfunkel were arguing and about to go their separate ways. Garfunkel decided to have a go at acting, he appears in the movie *Catch 22.* Simon is wishing him the best for his part in the movie Garfunkel is filming in Mexico.

As is usual with Paul Simon, the song’s lyrics are exquisite.

Tom, get your plane right on time

I know your part’ll go fine

Fly down to Mexico

Da-n-da-da-n-da-n-da-da a

And here I am

The only living boy in New YorkI get the news I need on the weather report

Oh, I can gather all the news I need on the weather report

Hey, I’ve got nothing to do today but smile

Da-n-do-da-n-do-da-n-do

Here I am

The only living boy in New YorkHalf of the time we’re gone but we don’t know where

And we don’t know where

Here I am

Half of the time we’re gone, but we don’t know where

And we don’t know whereTom, get your plane right on time

I know that you’ve been eager to fly now

Hey, let your honesty shine, shine, shine

Like it shines on me

The only living boy in New York

The only living boy in New York

Here is a video of this beautiful song. Enjoy!

]]>It has been a long process of several decades to understand the origin of the elements. In fact, we have not totally finished understanding the processes yet. But, we do know the story for most elements. All the hydrogen in the Universe was formed in the big bang. This is true for nearly all the helium too. A small amount of the 25% or so of helium in the Universe has been created within stars through the conversion of hydrogen into helium. But, not much has been created this way because most of that helium is further converted to carbon.

The only other element to be formed in the big bang is lithium. About 20% of the lithium in the Universe was formed in the big bang, the rest has been formed since,

Together, hydrogen and helium comprise 99% of the elements in the Universe by number (not by mass).

I have decided to use this fascinating table as the basis for a series of blogs over the next few weeks to explain each of the 6 processes in these six boxes

]]>*“Forever Young”* was recorded by Dylan in November 1973. The slow version runs for 4m57s and is the 6th track on *Planet Waves*, the last track on the first side of the record. The fast version (which is a shorter track at 2m49s) is the 7th track on the album, the first track on the second side of the record. Dylan first performed *“Forever Young”* live in January 1974 and his most recent live performance of it was in November 2011. He has performed it live a remarkable 493 times as of my writing this.

You may be familiar with a 1988 Rod Stewart song by the same name. Confusingly, it is not a cover version in the traditional sense, but bears such a remarkable similarity to Dylan’s song in both melody and some of the lyrics that Stewart agreed to share his royalties with Dylan (presumably to avoid a lawsuit).

The inspiration for the song was Dylan’s eldest son Jesse who was born in 1966. Dylan wrote *“Forever Young”* as a lullaby to his young son, and over the years it has been covered by many artists.

May God bless and keep you always

May your wishes all come true

May you always do for others

And let others do for you

May you build a ladder to the stars

And climb on every rung

May you stay forever young

Forever young, forever young

May you stay forever youngMay you grow up to be righteous

May you grow up to be true

May you always know the truth

And see the lights surrounding you

May you always be courageous

Stand upright and be strong

May you stay forever young

Forever young, forever young

May you stay forever youngMay your hands always be busy

May your feet always be swift

May you have a strong foundation

When the winds of changes shift

May your heart always be joyful

May your song always be sung

May you stay forever young

Forever young, forever young

May you stay forever young

Here is the official Vevo video of this great song. Enjoy!

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

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

Allowing us to re-write Equation (1) as

Remember that momentum is defined as

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*).

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

Assuming that both can change, we can therefore write

This allows us to write Equ. (2) as

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

Using the chain rule to differentiate this, we have

But, we can write

as

This allows us to write Equ. (5) as

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

Which is

So we can write

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

So

Integrating this we get

So

This tells us that an object has rest mass energy and that its total energy is given by

where is the *relativistic mass*.

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

where is the force applied, is the mass, and 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 . This raises the conundrum of why can’t we keep applying a force to a body of mass , 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

Where is the so-called *Lorentz factor* and 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

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

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.

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 moving relative to each other with a velocity in the x-direction. Also, the balls are going to have the same rest mass, , as measured in their respective frames and (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).

The blue ball moves solely in the y-direction in reference frame , and the red ball moves solely in the y-direction in reference frame . Ball starts by moving in the positive y-direction in reference frame with a velocity , and ball starts moving in the negative y-direction in reference frame with a velocity in frame .

Reference frame is moving relative to frame at a velocity in the positive x-direction. So, as seen in , the motion of ball appears as shown in the left of Figure 2. That is, it appears in 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 , ball will move vertically downwards in the negative y-direction, with a velocity . Ball 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 the motions of balls and looks like the diagram on the right of Figure 1. In , it is ball which moves vertically, and ball which moves in both the and directions.

To calculate the velocity of ball as seen in , 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 in which is moving relative to with a velocity , then the velocity in frame is given by

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

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

This time we write

and

So

Dividing each term in the right-hand side by , we get

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

After the collision, the velocity of ball becomes . What about ball ?

We can see that will not change, and after the collision will be .

We are now going to look at the momentum of balls and before and after the collision, as seen in frame . We will start off by assuming that the mass is constant for both balls, that is that 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

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

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

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

.

If we assume that momentum is conserved, we can write

So, if we assume that the mass of both ball and ball in frame is , the momentum in the y-direction is only conserved if . But, is only equal to unity when the relative velocity between the two frames is zero; in other words when the two frames are not moving relative to each other! If 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 needs to be greater than the mass of ball as seen in frame , as the speed of ball in the y-direction in frame .

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 , we have to allow the masses to change.

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

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

Where is the mass of ball in frame which is affected by its velocity in frame , which is .

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

Where , the Lorentz factor due to the relative velocity between and .

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

And, for ball we can write

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

For ball , we will write

where

(that is, depends on the speed of ball in frame , and that speed is ).

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

For ball , we will write

Where depends on the speed of ball as seen in frame . (**Note:** the mass does not depend on just the y-component of ball ‘s speed (as is often incorrectly stated), *it depends on its total speed*).

To calculate the value of we note that it is made up of the x-component and the y-component . But, , and we showed above that , where this .

Using Pythagoras to calculate , we have

so

Using this value of we can write

But, the terms can be factorised as

And so we can write

But, , so we can write

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

Comparing this to Equ. (7), the equation for , 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

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

]]>But, let us suppose that we have two reference frames and moving relative to each other with a speed of , 90% of the speed of light. Surely, if someone in frame fires a high-speed bullet at a speed of , an observer in frame will think that the bullet is moving away from him at a speed of , 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

To calculate the velocity of some object moving with a velocity in reference frame we need to use these Lorentz transformations.

We start off by writing

and

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

and

We can now write (the velocity of the object as seen in frame ) as

The terms cancel, and dividing each term on the right hand side by gives

where was the velocity of the object in reference frame .

Going back to our example of and , we can see that the velocity as measured by an observer in reference frame will be

, *not* as we naively calculated.

What happens if a person in reference frame shines a light in the same direction as is moving away from ? In this case, . Putting this into our equation for we get

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

]]>