Troll Maths:

Consider the sum of all natural number powers of 2, that is 2 + 4 + 8 + 16 + ……. up to infinity and call the total S.

Now S/2 = 1 + 2 + 4 + 8 +…. . So S = S/2 -1 and S = -2.

Therefore the sum of all the natural number powers of 2 is -2.

Problem?

The real problem with superinjuctions is that their £50k price tag makes them available only to the very wealthy. Regardless of whether or not a superinjuction can be justified, a legal measure usable by less than 1% of the population cannot be true equality in the eyes of the law and any reform of privacy law should take this into account. Imogen Thomas couldn’t afford £50k. I’m sure Mary Bale (she put a cat in a bin) would have taken out a superinjunction if she could have afforded it.

To  elaborate on the invariance of distance in 4d spacetime:

As I have previously shown, observers will disagree on distances in space and time depending on their inertial frame of reference. I will now demonstrate that we agree on the distance travelled by you on your bike to the shops as in the previous example.

It simplifies matters if we imagine a simple universe consisting of 1 spatial and 1 temporal dimension. Imagine that within our own inertial frame of reference we are spatially at rest but travelling at the speed of light along the time axis. The 1 spatial dimension happens to be a straight line that passes through your house and through the shops you are travelling to and on to infinity in either dimension. We will set the distance from your house to the shops as 1km in the positive direction.

Now we can calculate how far you travel in space and time on your journey according to your inertial frame of reference and mine.

It may seem counter-intuitive but you remain at rest within your own inertial frame of reference (you do not move any further away from yourself!) and you move along the time axis at the speed of light for 360 seconds. Note: there is a very tiny level of time and space dilation due to you moving relative to the span between your house and the shops, it will shorten by a fraction of a millimetre and you will gain a very tiny amount of time but we can ignore this for our calculations. Therefore the distance that you travel in spacetime according to you is the speed of light * 360 seconds = 107,925,284,880 meters.

Now imagine that my spaceship starts at the same point in space and time as you and moves in the positive spatial direction. According to me, moving away from you at 0.8 times the speed of light, you are moving in the negative spatial direction. To work out exactly how quickly we again use the formula V = V0 sqrt(1 - v^2/c^2) where V is my speed relative to C, V0 is my speed according to me, and c is the speed of light. We know I am moving at 0.8c so we can rearrange this formula to solve for V0. The surprising result is that according to me, I am travelling at 4/3c, faster than the speed of light! This may seem impossible but in reality I am not travelling faster than the speed of light, my perception that I am is due to time slowing down for me on board my ship. (Interestingly, the point at which one would think one was travelling at *exactly* the speed of light is c/sqrt2). Now that we know how fast you seem to be travelling away from me, we can work out how far you seem to travel to me in space and time. The distance that you travel is 216seconds * -4/3c (as you are moving in the negative direction according to me) – 600m (the distance that you travel according to me to reach the shops). This comes out at -86,340,227,304m. To calculate the distance in the temporal dimension we multiply c by 216 seconds to get 64,755,170,928m.

Now, these seem to wildly disagree with each other on the distance and time that you have travelled. By combining space and time into a 2 dimensional plane we can solve for the distance Z by using Pythagoras’ theorem Z^2 = X^2 + Y^2. Plugging in the numbers we get Z = sqrt(-86,340,227,304^2 + 64,755,170,928^2) =  107,925,284,880m, the same as the result we had previously obtained. Therefore we now agree on the distance in spacetime that you have travelled!

I have illustrated this with the diagram accompanying this post.

In this diagram we see that despite our differing perspectives on the time and space that you have travelled through that we both agree on the spacetime distance that you have travelled. In addition, anyone moving along the 1-dimensional universe that we have imagined in any direction and with any velocity will see the time and distance that you have travelled as somewhere on the semicircle. Therefore anyone in the universe can agree on the distance you have travelled!

RIP Brian Haw, 07/01/1949 - 18/06/2011

Brian Haw, the 10 year veteran protester has died of lung cancer while receiving treatment in Germany. I never really knew him but I’m honoured to have met him on a number of occasions. My most vivid memory of him is having a fag and a chat with him on a miserable, rainy new year’s day a few years ago. He was a very intense, troubled man who nevertheless had a lot of time for people. There was a sense of humour hidden deep down but the years of campaigning seemed to have put up a barrier against even allowing himself to smile. I have a massive amount of respect that he would give up everything for what he believed in. It’s a sad state of affairs when Nobel Peace Prize laureate Barack Obama is continuing two wars while dipping his toes in Libya, keeping Guantanamo bay open and supporting the torture of Bradley Manning while Brian Haw, a man who deserved the prize more than Obama ever will, only ever received ham-fisted legal threats and attempts to silence him. The UK Government never had a legal or moral case for removing him. In the end only death could stop this heroic, courageous man.

“On June 2 2001, the police came along and said: ‘How long you going to be here, Brian?’ I said: ‘As long as it takes.’”

Image by David Hunt, via Wikimedia commons http://bit.ly/kr5aMM

The most touching and fun love song I’ve heard this year. 

The whole “Will AV help or hinder the BNP” debate is an irrelevant distraction from the discussion about actual issues. The question that we should be asking and pushing is “Does AV more accurately represent the will of the electorate as a whole than the current system” to which I believe the answer is yes. Living in a democracy means people have the right to vote for candidates one may not like. The BNP have no chance of winning a seat under either system at any rate.

Someone on Twitter asked me to explain why in Physics space and time are combined into “spacetime”. While this isn’t a complete answer (I’d have to write a book) I think this at least hints at it. My answer was as follows:

To understand why we combine space and time to form a 4-dimensional “spacetime” we must first understand that in Einstein’s special relativity that time and space are not absolute; they depend of the observer’s intertial frame of reference.

Note: I won’t show the derivations or otherwise explain the formulae that I use. The mathematics can get a little convoluted and it isn’t really necessary.

Imagine that you are travelling to the shop on your bicycle. The shops are 1km away, you travel at a constant speed of 10km per hour and stop once you reach the shops. Therefore according to you, you travel 1000m in 360 seconds. Now, imagine that I am flying over the Earth in my spaceship at 4/5 the speed of light.

The distance that you travel is given by L = L0 sqrt(1-(v^2)/(c^2)) where L is the distance you travel according to me, L0 is the distance you travel according to you, v is the speed I am travelling at and c is the speed of light.

Plugging the numbers in we get L = 1000 sqrt (1 – (0.8)^2) = 600m.

Likewise, time depends on the inertial frame of reference of the observer, and is given by the equation T = T0 (sqrt(1-(v^2)/(c^2)). Where T is the time you take according to me, T0 is the time you take according to you, v is my speed and c is the speed of light.

Again, plugging the numbers in we get T = 360 *0.6 = 216 seconds.

Therefore to recap: According to you, you have travelled 1000m in 360 seconds but according to me you have travelled 600m in 216 seconds. We disagree on the distance you have travelled as well as the time that you have taken. Following on from this it can be shown that time and space aren’t absolute. I havn’t really proven this so far but this should give you an idea of why they aren’t absolute.

By combining space and time together into a 4d “spacetime” where distance is given by: S^2 = a^2 + b^2 + c^2 - t^2 where a, b and c are the distance components travelled in each dimension of 3d space and t is the time component. (this may look familiar! It is similar to Pythagoras’ theorem but scaled up to 4d hyperbolic “Minkowski” space rather than in 2d or 3d “Euclidean” space). Using this formula we both agree on the distance that you have travelled in 4d spacetime.

There are other issues involved, such as issues with causality brought up by special relativity that combining space and time resolve but explaining these would take far longer and require a far better teacher than myself.

Okay Banksy if you’re reading here’s 2 freebies.

1: This is what democracy looks like: (picture of Simon Cowell); This is what democracy sounds like: (picture of Jedward).

2: Mcdonalds logo with “in hoc signo vinces” painted across it in US military stencil style text. Easy.

Street Art by Banksy http://www.banksy.co.uk/ 2010 in Dumbo, New York