The Mathematics of Lightspeed

There’s a whole range of other theories that could explain this. A stake is one of the greatest prizes of science, a grand theory of everything. The first issue is to consider whether the speed of light is really the absolute barrier that Einstein described. Professor Marcus du Sautoy looks at the mathematics of lightspeed.

There are at least two arguments that suggest it might be possible, in certain circumstances, to travel faster than the speed of light. The intriguing thing is that, mathematically speaking , travelling faster than the speed of light isn’t quite as difficult as the popular interpretation of Einstein’s theory suggests. In fact, from a mathematical point of view it isn’t impossible at all and to understand why you need to explore the relationship between physics and mathematics.

There are many examples in the history of physics where mathematics predicts something that, at first sight, is counter-intuitive only for the mathematics then to be proved right.

Paul Dirac

Back in the 1920s, a scientist called Paul Dirac came up with equations to describe what happened to electrons when they travelled close to the speed of light. But, his equations led to a peculiar conclusion. They predicted that every particle had an equivalent anti-particle with an opposite electric charge. These anti-particles were combine to form anti-matter. At the time, the idea of anti-matter seemed mad, but eventually, incontrovertible evidence for its existence was found.

We’ve seen something similar happen with the prediction that neutrinos would exist, before they’d been observed. So, maths can sometimes suggest solutions that appear impossible in the real world but, then turn out to be feasible after all.

Absolute Barrier

Surprisingly enough there are mathematical solutions to Einstein’s equations which do allow particles to go faster than the speed of light. We even have a name to describe these theoretical particles that can do this - tachyons.

On the surface tachyons are pretty strange, most notably there mass is an imaginary number . But, however strange that sounds it doesn’t mean they couldn’t exist. A large part of the universe is built on imaginary numbers. So, what’s special about tachyons, how could they travel faster than the speed of light?

The key is this, Einstein’s formula forbids any particle to travel through the speed of light because as it accelerates, its mass gets greater and greater, but if a particle is formed when its already travelling beyond the speed of light, it gets passed this problem.

Even before these results, a few scientists had suggested that neutrinos might have a tachyonic behaviour . In other words there might be a link between tachyons and neutrinos. It is too early to say whether this theory has any legs, but it’s still good to know that from a mathematical perspective that it is possible to travel faster than the speed of light.

There’s another reason for doubting that Einstein’s speed limit is quite as absolute as it appears. In fact, there are certain circumstances where the idea of an ultimate speed limit doesn’t make any sense.

The exciting thing for me about controversial results like these are they shake things up, they provoke lots of questions, demand new ideas and in soing so shine a light on theoretical problems that tend to get swept under the carpet. Unless you study science you could be forgiven for thinking that the theories used by academics to describe the universe all join up nicely , but that’s not always the case.

Professor Joao Magueijo

Professor Joao Magueijo “Obviously this result contradicts what you find in textbooks but, if you’re working in the frontier of physics trying to find new theories this is not as tragic as you might think. Its a crisis but, we need a crisis because a lot of things in physics, in those textbooks make no sense.”

Einstein’s theories describe with astonishing accuracy the universe we can see. The planets, the stars, even the distant galaxies and here the speed of light is indeed the ultimate speed limit, but even within this familiar universe there are places Einstein’s theories don’t work. In extreme conditions, the rules break down. Physics hasn’t yet developed the language to understand. what happened inside a black hole, for example Einstein’s speed limit also causes problems when trying to explain how the universe evolved from the birth of everything - the Big Bang.

Physicists think that at the moment of the Big Bang everything in the universe was crammed into one tiny point smaller than an atom. At the Big Bang the universe expanded at astonishing speed. As it expanded it cooled allowing fundamental particle then protons and neutrons to condense out of the energetic soup. All of this happened in less than a second.

Over the next 400,000 years the universe cooled enough to allow the first hydrogen atoms to form, creating vast clouds of gas that finally began to collapse into the familiar stars and galaxies that make up the universe as we see it today.

But, here’s the big problem: accepted science only seems to account for what happened just after the Big Bang. If you want to understand what happened to our universe in its very first moments Einstein can’t help you. and there’s a particular problem with Einstein’s idea of a constant cosmic speed limit when you apply it to the Big Bang.

Some physicists believe that for the universe around us to be as we see it today then that speed limit must have been broken in the instants immediately after the Big Bang.