# Black Holes and the Information Paradox: a Solution?

Gravity has a good sense of humour. On the one hand, it’s the weakest force we know. The upward push of your chair is more than enough to counteract the pull of the entire planet! Yet gravity has an ace up its sleeve – unlike all other forces, it’s always attractive. For larger objects, the other forces start bickering and cancelling out. But gravity just keeps on getting stronger, until it’s impossible to escape – a black hole!

As a theoretical physicist, I tend to carry a black hole whenever I’m travelling.

As you can see, the top of this bucket is the surface of a black hole, otherwise known as the event horizon. When I release water from a glass above the black hole, it is attracted to the black hole, and falls inexorably towards it, never to be seen again.

Okay, I suppose my bucket isn’t a real black hole. After all, it’s the gravity of the Earth that pulls the water in. And light can definitely escape because I can see inside it! But it does accurately represent the bending of space and time. Albert Einstein taught us that everything in the universe rolls around on the cosmic quilt of spacetime, like balls on an elastic sheet. A heavy ball distorts the sheet, creating my black hole bucket.

You might not feel too threatened by black holes – after all, the nearest one is probably 8 billion billion miles away. But in actual fact you could be falling into a black hole right now without noticing! Turns out that for a large enough black hole, the event horizon is so far away that gravity there is very weak. So there’s no reason why you should experience anything special.

This disturbing fact has an unexpected consequence from the microscopic world of quantum mechanics. Every quantum theory must have a single vacuum, essentially the most boring and lazy state of affairs. If I stand still the vacuum is just empty space. But as soon as I start accelerating something weird happens. Particles suddenly appear from nowhere!

What does that mean for our black hole? Well if you’re not falling in, you must be accelerating away to oppose the huge pull of gravity! This means that you should see the black hole glowing with particles called Hawking radiation. Remember my black hole full of water? Well, you haven’t fallen in. And that means I have to cover you with Hawking radiation!

Luckily for your computer, the Hawking confetti that came out isn’t the same as the water that went in. From your perspective the water has simply disappeared! Exactly the same thing seems to happen for real black holes.

This black hole magic trick has become infamous among scientists, resisting all efforts at explanation. But a solution might be at hand, courtesy of Hawking himself! What if you could slightly change the vacuum every time something dropped into the black hole? Then, if you’re very careful, you might just be able to reconstruct the original water from the confetti of Hawking radiation.

Put another way, the event horizon takes a lock of soft hair from every passing particle as a memento of its existence. This information is eventually carried off by Hawking’s magic particles, reminding us of what we’d lost. It remains for soft experts, like myself, to work out the exact details.

This post is based on a talk given for the Famelab competition. You can read the full paper by Stephen Hawking, Malcolm Perry and Andy Strominger here.

# Romeo and Juliet, through a Wormhole

I spent last week at the Perimeter Institute in Waterloo, Ontario. Undoubtedly one of the highlights was Juan Maldenena’s keynote on resolving black hole paradoxes using wormholes. Matt’s review of the talk below is well worth a read!

Perimeter is hosting this year’s Mathematica Summer School on Theoretical Physics. The school is a mix of lectures on a topic in physics (this year, the phenomenon of quantum entanglement) and tips and tricks for using the symbolic calculation program Mathematica.

Juan Maldacena is one of the lecturers, which gave me a chance to hear his Romeo and Juliet-based explanation of the properties of wormholes. While I’ve criticized some of Maldacena’s science popularization work in the past, this one is pretty solid, so I thought I’d share it with you guys.

You probably think of wormholes as “shortcuts” to travel between two widely separated places. As it turns out, this isn’t really accurate: while “normal” wormholes do connect distant locations, they don’t do it in a way that allows astronauts to travel between them, Interstellar-style. This can be illustrated with something called a Penrose…

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# T-duality and Isometries of Spacetime

I’ve just been to an excellent seminar on Double Field Theory by its co-creator, Chris Hull. You may know that string theory exhibits a meta-symmetry called T-duality. More precisely, it’s equivalent to put closed strings on circles of radius $R$ and $1/R$.

This is the simplest version of T-duality, when spacetime has no background fields. Now suppose we turn on the Kalb-Ramond field $B$. This is just an excitation of the string which generalizes electromagnetic potential.

This has the effect of making T-duality more complicated. In fact it promotes the $R\to 1/R$ symmetry to $O(d,d;\mathbb{Z})$ where $d$ is the dimension of your torus. Importantly for this to work, we must choose a $B$ field which is constant in the compact directions, otherwise we lose the isometries that gave us T-duality in the first place.

Under this T-duality, the $B$ field and $G$ metric get mixed up. This can have dramatic consequences for the underlying geometry! In particular our new metric may not patch together by diffeomorphisms on our spacetime. Similarly our new Kalb-Ramond field $B$ may not patch together via diffeomorphisms and gauge transformations. We call such strange backgrounds non-geometric.

To express this more succintly, let’s package diffeomorphisms and gauge transformations together under the name generalized diffeomorphisms. We can now say that T-duality does not respect the patching conditions of generalized diffeomorphisms. Put another way, the $O(d,d)$ group does not embed within the group of generalized diffeomorphisms of our spacetime!

This lack of geometry is rather irritating. We physicists tend to like to picture things, and T-duality has just ruined our intuition! But here’s where Double Field Theory comes in. The idea is to double the coordinates of your compact space, so that $O(d,d)$ transformations just become rotations! Now T-duality clearly embeds within generalized diffeomorphisms and geometry has returned.

All this complexity got me thinking about an easier problem – what do we mean by an isometry in a theory with background fields? In vacuum isometries are defined as diffeomorphisms which preserve the metric. Infinitesimally these are generated by Killing vector fields, defined to obey the equation

$\displaystyle \mathcal{L}_K g=0$

Now suppose you add in background fields, in the form of an energy-momentum tensor $T$. If we want a Killing vector $K$ to generate an overall symmetry then we’d better have

$\displaystyle \mathcal{L}_K T=0$

In fact this equation follows from the last one through Einstein’s equations. If your metric solves gravity with background fields, then any isometry of the metric automatically preserves the energy momentum tensor. This is known as the matter collineation theorem.

But hang on, the energy momentum tensor doesn’t capture all the dynamics of a background field. Working with a Kalb-Ramond field for instance, it’s the potential $B$ which is the important quantity. So if we want our Killing vector field to be a symmetry of the full system we must also have

$\displaystyle \mathcal{L}_K B=0$

at least up to a gauge transformation of $B$. Visually if we have a magnetic field pointing upwards everywhere then our symmetry diffeomorphism had better not twist it round!

So from a physical perspective, we should really view background fields as an integral part of spacetime geometry. It’s then natural to combine fields with the metric to create a generalized metric. A cute observation perhaps, but it’s not immediately useful!

Here’s where T-duality joins the party. The extended objects of string theory (and their low energy descriptions in supergravity) possess duality symmetries which exchange pieces of the generalized metric. So in a stringy world it’s simplest to work with the generalized metric as a whole.

And that brings us full circle. Double Field Theory exactly manifests the duality symmetries of the generalized metric! Not only is this mathematically helpful, it’s also an important conceptual step on the road to unification via strings. If that road exists.

# Gravity for Beginners

Everyone knows about gravity. Every time you drop a plate or trip on the stairs you’re painfully aware of it. It’s responsible for the thrills and spills at theme parks and the heady hysteria of a plummeting toboggan. But gravity is not merely restricted to us small fry on Earth. It is a truly universal force, clumping mass together to form stars and planets, keeping worlds in orbit around their Suns. Physicists call gravity a fundamental force – it cannot be explained in terms of any other interaction.

The most accurate theory of gravity is Einstein’s General Theory of Relativity. The presence of mass warps space and time, creating the physical effects we observe. The larger the mass, the more curved space and time become, so the greater the gravitational pull. Space and time are a rubber sheet, which a large body like the Sun distorts.

Gravity is very weak compared to other fundamental forces. This might be a bit of a surprise – after all it takes a very powerful rocket to leave Earth’s orbit. But this is just because Earth is so very huge. Things on an everyday scale don’t seem to be pulled together by gravity. But small magnets certainly are attracted to each other by magnetism. So magnetism is stronger than gravity.

The fact that gravity acts by changing the geometry of space and time sets it apart from all other forces. In fact our best theories of particles assume that there is some cosmic blank canvas on which interactions happen. This dichotomy and the weakness of gravity give rise to the conflicts at the heart of physics.

It’s All Relative, After All

“You can’t just let nature run wild.”

Walt Disney

Throughout history, scientists have employed principles of relativity to understand nature. In broad terms these say that different people doing the same scientific experiment should get the same answer. This stands to reason – we travel the world and experience the same laws of nature everywhere. Any differences we might think we observe can be explained by a change in our experimental conditions. Looking for polar bears in the Sahara is obvious madness.

In the early 17th century Galileo formulated a specific version of relativity that applied to physics. He said that the laws of physics are the same for everyone moving at constant speed, regardless of the speed they are going. We notice this in everyday life. If you do an egg and spoon race walking at a steady pace the egg will stay on the spoon just as if you were standing still.

If you try to speed up or slow down, though, the egg will likely go crashing to the ground. This shows that for an accelerating observer the laws of physics are not the same. Newton noticed this and posited that an accelerating object feels a force that grows with its mass and acceleration. Sitting in a plane on takeoff this force pushes you backwards into your seat. Physicists call this Newton’s Second Law.

Newton’s Second Law can be used the opposite way round too. Let’s take an example. Suppose we drop an orange in the kitchen. As it travels through the air its mass certainly stays the same and the only significant force it feels is gravity. Using Newton’s Second Law we can calculate the acceleration of the orange. Now acceleration is just change in speed over time. So given any time in the future we can predict the speed of the orange.

But speed is change in distance over time. So we know the distance the orange has travelled towards the ground after any given time. In particular we can say exactly how much time it’ll be before the orange goes splat on the kitchen floor. One of the remarkable features of the human brain is that it can do approximations to these calculations very quickly, enabling us to catch the orange before disaster strikes.

The power of the principle of relativity is now apparent. Suppose we drop the orange in a lift, while steadily travelling upwards. We can instantly calculate how long it’ll take to hit the lift floor. Indeed by the principle of relativity it must take exactly the same amount of time as it did when we were in the kitchen.

There’s a hidden subtlety here. We’ve secretly assumed that there is some kind of universal clock, ticking away behind the scenes. In other words, everyone measures time the same, no matter how they’re moving. There’s also a mysterious cosmic tape measure somewhere offstage. That is, everyone agrees on distances, regardless of their motion. These hypotheses are seemingly valid for everyday life.

But somehow these notions of absolute space and time are a little unsettling. It would seem that Galileo’s relativity principle applies not only to physics, but also to all of space and time. Newton’s ideas force the universe to exist against the fixed backdrop of graph paper. Quite why the clock ticks and the ruler measures precisely as they do is not up for discussion. And the mysteries only deepen with Newton’s theory of gravity.

A Tale of Two Forces

“That one body may act upon another at a distance through a vacuum without the mediation of anything else […] is to me so great an absurdity that […] no man who has […] a competent faculty of thinking could ever fall into it.”

Sir Isaac Newton

Newton was arguably the first man to formulate a consistent theory of gravitation. He claimed that two masses attract each other with a force related to their masses and separation. The heavier the objects and the closer they are, the bigger the force of gravity.

This description of gravity was astoundingly successful.  It successively accounted for the curvature of the Earth, explained the motion of the planets around the Sun, and predicted the precise time of appearance of comets. Contemporary tests of Newton’s theory returned a single verdict – it was right.

Nevertheless Newton was troubled by his theory. According to his calculations, changes in the gravitational force must be propagated instantaneously throughout the universe. Naturally he sought a mechanism for such a phenomenon. Surely something must carry this force and effect its changes.

Cause and effect is an ubiquitous feature of everyday physics. Indeed Newton’s Second Law says motion and force and inextricably linked. Consider the force which opens a door. It has a cause – our hand pushing against the wood – and an effect – the door swinging open. But Newton couldn’t come up with an analogous explanation for gravity. He had solved the “what”, but the “why” and “how” evaded him entirely.

For more than a century Newton’s theories reigned supreme. It was not until the early 1800s that physicists turned their attention firmly towards another mystery – electricity. Michael Faraday lead the charge with his 1721 invention of an electric motor. By placing a coil of wire in a magnetic field and connecting a battery he could make it rotate. The race was on to explain this curious phenomenon.

Faraday’s work implied that electricity and magnetism were two sides of the same coin. The strange force he had observed became known, appropriately, as electromagnetism. The scientific community quickly settled on an idea. Electricity and magnetism were examples of a force field – at every point in space surrounding a magnet or current, there were invisible lines of force which affected the motion of nearby objects.

All they needed now were some equations. This would allow them to predict the behaviour of currents near magnets, and verify that this revolutionary force field idea was correct. Without a firm mathematical footing the theory was worthless. Several preeminent figures tried their hand at deriving a complete description, but in 1861 the problem remained open.

In that year a young Scotsman named James Clerk Maxwell finally cracked the issue. By modifying the findings of those before him he arrived at a set of equations which completely described electromagnetism. He even went one better than Newton had with gravity. He found a mechanism for the transmission of electromagnetic energy.

Surprisingly, Maxwell’s equations suggested that light was an electromagnetic wave. To wit, solutions showed that electricity and magnetism could spread out in a wavelike manner. Moreover the speed of these waves was determined by a constant in his equations. This constant turned out to be very close to the expected speed of light in a vacuum. It wasn’t a giant leap to suppose this wave was light itself.

At first glance it seems like these theories solve all of physics. To learn about gravity and the motion of uncharged objects we use Newton’s theory. To predict electromagnetic phenomena we use Maxwell’s. Presumably to understand the motion of magnets under gravity we need a bit of both. But trying to get Maxwell’s equations to play nice with Galileo’s relativity throws a big spanner in the works.

A Breath of Fresh Air

“So the darkness shall be the light, and the stillness the dancing.”

T.S. Eliot

Galileo’s relativity provides a solid bedrock for Newtonian physics. It renders relative speeds completely irrelevant, allowing us to concentrate on the effects of acceleration and force. Newton’s mechanics and Galileo’s philosophy reinforce each other. In fact Newton’s equations look the same no matter what speed you are travelling at. Physicists say they are invariant for Galileo’s relativity.

We might naively hope that Maxwell’s equations are invariant. Indeed a magnet behaves the same whether you are sitting still at home or running around looking for buried treasure. Currents seem unaffected by how fast you are travelling – an iPod still works on a train. It would be convenient if electromagnetism had the same equations everywhere.

Physicists initially shared this hope. But it became immediately apparent that Maxwell’s equations were not invariant. A change in the speed you were travelling caused a change in the equations. Plus there was only one speed at which solutions to the equations gave the right answer! Things weren’t looking good for Galileo’s ideology.

To solve this paradox, physicists made clever use of a simple concept. Since antiquity we’ve had a sense of perspective – the world looks different from another point of view. Here’s a simple example in physics. Imagine you’re running away from a statue. From your viewpoint the statue is moving backwards. From the statue’s viewpoint you are moving forwards. Both descriptions are right. They merely describe the same motion in contrary ways.

In physics we give perspective a special name. Any observer has a frame of reference from which they see the world. In your frame of reference the statue moves backwards. In the statue’s frame of reference you move forwards. We’ve seen that Newton’s physics is the same in every frame of reference which moves with a constant speed.

We can now rephrase our discovery about Maxwell’s equations. Physicists found that there was one frame of reference in which they were correct. Maxwell’s equations somehow prefer this frame of reference over all other ones! Any calculations in electromagnetism must be done relative to this fixed frame. But why is this frame singled out?

Faced with this question, physicists spotted the chance to kill two birds with one stone. The discovery that light is a wave of electromagnetism raises an immediate question. What medium carries the light waves? We’re used to waves travelling through some definite substance. Water waves need a sea or ocean, sound waves require air, and earthquakes move through rock. Light waves must travel through a fixed mysterious fog pervading all of space. It was called aether.

The aether naturally has it’s own frame of reference, just as you and I do. When we measure the speed of light in vacuum, we’re really measuring it relative to the aether. So the aether is a special reference frame for light. But light is an electromagnetic wave. It’s quite sensible to suggest that Maxwell’s preferred reference frame is precisely the aether!

Phew, we’ve sorted it. Newtonian mechanics works with Galilean relativity because there’s nothing to specify a particular reference frame. Maxwell’s equations don’t follow relativity because light waves exist in the aether, which is a special frame of reference. Once we’ve found good evidence for the aether we’ll be home and dry.

So thought physicists in the late 19th century. The gauntlet was down. Provide reliable experimental proof and win instant fame. Two ambitious men took up the challenge – Albert Michelson and Edward Morley.

They reasoned that the Earth must be moving relative to the aether. Therefore from the perspective of Earth the aether must move. Just as sound moves faster when carried by a gust of wind, light must move faster when carried by a gust of aether. This prediction became known as the aether wind.

By measuring the speed of light in different directions, Michelson and Morley could determine the direction and strength of the aether wind. The speed of light would be greatest when they aligned their measuring apparatus with the wind. Indeed the gusts of aether would carry light more swiftly. The speed of light would be slowest when they aligned their apparatus at right angles to the wind.

The expected changes in speed were so minute that it required great ingenuity to measure them. Nevertheless by 1887 they had perfected a cunning technique. With physicists waiting eagerly for confirmation, Michelson and Morley’s experiment failed spectacularly. They measured no aether wind at all.

The experiment was repeated hundreds of times in the subsequent years. The results were conclusive. It didn’t matter which orientation you chose, the speed of light was the same. This devastating realisation blew aether theory to smithereens. Try as they might, no man nor woman could paper over the cracks. Physics needed a revolution.

How Time Flies

“Put your hand on a hot stove for a minute, and it seems like an hour. Sit with a pretty girl for an hour, and it seems like a minute. That’s relativity.”

Albert Einstein

It is the most romanticised of fables. The task of setting physics right again fell to a little known patent clerk in Bern. The life and work of Albert Einstein has become a paradigm for genius. But yet the idea that sparked his reworking of reality was beautifully simple.

For a generation, physicists had been struggling to reconcile Galileo and Maxwell. Einstein claimed that they had missed the point. Galileo and Maxwell are both right. We’ve just misunderstood the very nature of space and time.

This seems a ridiculously bold assumption. It’s best appreciated by way of an analogy. Suppose you live in a rickety old house and buy a nice new chair. Placing it in your lounge you notice that it wobbles slightly. You wedge a newspaper under one leg. At first glance you’ve solved the problem. But when you sit on the chair it starts wobbling again.

After a few more abortive attempts with newspapers, rags and other household items you decide to put the chair in another room. But the wobble won’t go away. Even when you sand down the legs you can’t make it stand firm. The chair and the house seem completely incompatible.

One day a rogue builder turns up. He promises to fix your problem. You don’t believe he can. Neither changing the house nor the chair has made the slightest difference to you. When you return from work you are aghast to see he’s knocked the whole house down. He shouts up to you from a hole in the ground, “just fixing your foundations”!

More concretely Einstein said that there’s no problem with Galileo’s relativity. The laws of physics really are the same in every frame moving at constant speed. Physicists often rename this idea Einstein’s special principle of relativity even though it is Galileo’s invention! Einstein also had no beef with Maxwell’s equations. In particular they are the same in every frame.

To get around the fact that Maxwell’s equations are not invariant for Galileo’s relativity, Einstein claimed that we don’t understand space and time. He claimed that there is no universal tape measure, or cosmic timepiece. Everybody is responsible for their own measurements of space and time. This clears up some of the issues that annoyed Newton.

In order to get Maxwell’s equations to be invariant when we change perspective, Einstein had to alter the foundations of physics. He postulated that each person’s measurements of space and time were different, depending on how fast they were going. This correction magically made Maxwell’s equations work with his principle of special relativity.

There’s an easy way to understand how Einstein modified space and time. We’re used to thinking that we move at a constant speed through time. Clocks, timers and watches all attest to our obsession with measuring time consistently. The feeling of time dragging on or whizzing by is merely a psychological curiosity. Before Einstein space didn’t have this privilege. We only move through space as we choose. And moving through space has no effect on moving through time.

Einstein made everything much more symmetrical. He said that we are always moving through both space and time. We can’t do anything about it. Everyone always moves through space and time at the same speed – the speed of light. To move faster through space you must move slower through time to compensate. The slower you trudge through space, the faster you whizz through time. Simple as.

Einstein simply put space and time on a more even footing. We call the whole construct spacetime. Intuitively spacetime is four dimensional. That is, you can move in four independent directions. Three of these are in space, up-down, left-right, forward-backward. One of these is in time.

You probably haven’t noticed it yet, but special relativity has some weird effects. First and foremost, the speed of light is the same however fast you are going. This is because it is a constant in Maxwell’s equations, which are the same in all frames. You can never catch up with a beam of light! This is not something we are used to from everyday life.

Nevertheless it can be explained using Einstein’s special relativity. Suppose you measure the speed of light when you are stationery. You do this by measuring the amount of time it takes for light to go a certain distance. For the sake of argument assume you get the answer 10 mph.

Now imagine speeding up to 5 mph. You measure the speed of light again. Without Einstein you’d expect the result to be 5 mph. But because you’re moving faster through space you must be moving slower through time. That means it’ll take the light less time to go the same distance. In fact the warping of spacetime precisely accounts for the speed you’ve reached. The answer is again 10 mph.

Einstein’s spacetime also forces us to forget our usual notions of simultaneity. In everyday physics we can say with precision whether two events happen at the same time. But this concept relies precisely on Newton’s absolute time. Without his convenient divine timepiece we can’t talk about exact time. We can only work with relative perspectives.

Let’s take an example. Imagine watching a Harrison Ford thriller. He’s standing on the top of a train as it rushes through the station. He positions himself precisely in the center of the train. At either end is a rapier wielding bad guy eager to kill him. Ford is equipped with two guns which he fires simultaneously at his two nemeses. These guns fire beams of light that kill the men instantly.

In Ford’s frame both men die at the same time. The speed of light is the same in both directions and he’s equidistant from the men when he shoots. Therefore the beams hit at the same moment according to Ford. But for Ford’s sweetheart on the platform the story is different.

Let’s assume she is aligned with Ford at the moment he shoots. She sees the back of the train catching up with the point where Ford took the shot. Moreover she sees the front of the train moving away from the point of firing. Now the speed of light is constant in all directions for her. Therefore she’ll see the man at the back of the train get hit before the man at the front!

Remarkably all of these strange effects have been experimentally confirmed. Special relativity and spacetime really do describe our universe. But with our present understanding it seems that making genuine measurements would be nigh on impossible. We’ve only seen examples of things we can’t measure with certainty!

Thankfully there is a spacetime measurement all observers can agree on. This quantity is known as proper time. It’s very easy to calculate the proper time between events A and B. Take a clock and put it on a spaceship. Set your spaceship moving at a constant speed so that it goes through the spacetime points corresponding to A and B. The proper time between A and B is the time that elapses on the spaceship clock between A and B.

Everyone is forced to agree on the proper time between two events. After all it only depends on the motion of the spaceship. Observers moving at different speeds will all see the same time elapsed according to the spaceship clock. The speed that they are moving has absolutely no effect on the speed of your spaceship!

You might have spotted a potential flaw. What if somebody else sets off a spaceship which takes a different route from A to B? Wouldn’t it measure a different proper time? Indeed it would. But it turns out this situation is impossible. Remember that we had to set our spaceship off at a constant speed. This means it is going in a straight line through spacetime. It’s easy to see that there’s only one possible straight line route joining any two points. (Draw a picture)!

That’s it. You now understand special relativity. In just a few paragraphs we’ve made a huge conceptual leap. Forget about absolute space and time – it’s plain wrong. Instead use Einstein’s new magic measurement of proper time. Proper time doesn’t determine conventional time or length. Rather it tells us about distances in spacetime.

Einstein had cracked the biggest problem in physics. But he wasn’t done yet. Armed with his ideas about relativity and proper time, he turned to the Holy Grail. Could he go one better than Newton? It was time to explain gravity.

The Shape Of Space

“[…] the great questions about gravitation. Does it require time? […] Has it any reference to electricity? Or does it stand on the very foundation of matter – mass or inertia?”

James Clerk Maxwell

Physicists are always happy when they are going at constant speed. Thanks to Galileo and Einstein they don’t need to worry exactly how fast they are going. They can just observe the world and be sure that their observations are true interpretations of physics.

We can all empathise with this. It’s much more pleasant going on a calm cruise across the Aegean than a rocky boat crossing the English Channel. This is because the choppy waters cause the boat to accelerate from side to side. We’re no longer travelling at a constant speed, so the laws of physics appear unusual. This can have unpleasant consequences if we don’t find our sea legs.

Given this overwhelming evidence it seems madness to alter Einstein and Galileo’s relativity. But this is exactly what Einstein did. He postulated a general principle of relativity – the laws of physics are the same in any frame of reference. That is, no matter what your speed or acceleration.

To explain this brazen statement, we’ll take an example. Imagine you go to Disneyland and queue for the Freefall Ride. In this terrifying experience you are hoisted 60 metres in the air and then dropped. As you plunge towards the ground you notice that you can’t feel your weight. For a few moments you are completely weightless! In other words there is no gravity in your accelerating frame.

Let’s try a similar thought experiment. Suppose you wake up and find yourself in a bare room with no windows or doors. You stick to the floor as you usually would on Earth. You might be forgiven for thinking that you were under the influence of terrestrial gravity. In fact you could equally be trapped in a spaceship, accelerating at precisely the correct rate to emulate the force of gravity. Just think back to Newton’s Second Law and this is obviously true.

Hopefully you’ve spotted a pattern. Acceleration and gravity are bound together. In fact there seems to be no way of unpicking the knots. Immediately Einstein’s general relativity becomes more credible. An accelerating frame is precisely a constant speed frame with gravity around it.

So far so good. But we haven’t said anything about gravity that couldn’t be said about other forces. If we carry on thinking of gravity as a conventional interaction the argument becomes quite circular. We need a description of gravity that is independent of frames.

Here special relativity comes to our rescue. Remember that proper time gives us a universally agreed distance between two points. It does so by finding the length of a straight line in spacetime from one event to the other. This property characterises spacetime as flat.

To see this imagine you are a millipede, living on a piece of string. You can only move forwards or backwards. In other words you have one dimension of space. You also move through time. Therefore you exist in a two dimensional spacetime.

One day you decide to make a map of all spacetime. You travel to every point in spacetime and write down the proper time it took to get there. Naturally you are very efficient, and always travel by the shortest possible route. When you return you try to make a scale drawing of all your findings on a flat sheet of paper.

Suppose your spacetime is exactly as we’d expect from special relativity. Then you have no problem making your map. Indeed you always travelled by the shortest route to each point, which is a straight line in the spacetime of special relativity. On your flat sheet of paper, the shortest distance between two points is also a straight line. So your map will definitely work out.

You can now see why special relativistic spacetime is flat. Just for fun, let’s assume that you had trouble making your map. Just as you’ve drawn a few points on the map you realise there’s a point which doesn’t fit. Thinking you must be mistaken you try again. The same thing happens. You go out for another trek round spacetime. Exhausted you return with the same results. How immensely puzzling!

Eventually you start to realise that something fundamental is wrong. What if spacetime isn’t flat? You grab a nearby bowling ball and start drawing your map on its surface. Suddenly everything adds up. The distances you measured between each point work out perfectly. Your spacetime isn’t flat, it’s spherical.

Let’s look a bit closer at why this works. When you went out measuring your spacetime you took the shortest route to every point. On the surface of a sphere this is not a straight line. Rather it is part of a circle – a line of longitude or latitude. If you aren’t convinced, find a globe and measure it yourself on the Earth’s surface.

When you try to make a flat map, it doesn’t work. This is because the shortest distance between two points on the flat paper is a straight line. The two systems of measurement are just incompatible. We’re used to seeing this every time we look at a flat map of the world. The distances on it are all wrong because it has to be stretched and squished to fit the flat page.

Gallivanting millipedes aside, what has this got to do with reality? If we truly live in the universe of special relativity then we don’t need to worry about such complications. Everything is always flat! But Einstein realised that these curiosities were exactly the key to the treasure chest of gravity.

In his landmark paper of 1915, Einstein told us to forget gravity as a force. Instead, he claimed, gravity modifies the proper time between events. In doing so, it changes the geometry of spacetime. The flat, featureless landscape of special relativity was instantly replaced by cliffs and ravines.

More precisely Einstein came up with a series of equations which described how the presence of mass changes the calculation of proper time. Large masses can warp spacetime, causing planets to orbit their suns and stars to form galaxies. Nothing is immune to the change in geometry. Even light bends around massive stars. This has been verified countless times during solar eclipses.

Einstein had devised a frame independent theory of gravity which elegantly explained all gravitational phenomena. His general principle of relativity was more than vindicated; it became the title of a momentous new perspective on the cosmos. The quest to decipher Newton’s gravity was complete.

In the past century, despite huge quantities of research, we have made little progress beyond Einstein’s insights. The experimental evidence for general relativity is tumultuous. But yet it resists all efforts to express it as a quantum theory. We stand at an impasse, looking desperately for a way across.

# Affine Parameters and Euler-Lagrange Equations

Earlier today I was struggling to see why I couldn’t derive the general geodesic equation (*)

$\frac{\textrm{d}}{\textrm{d}\lambda}(g_{ab}\dot{x}^b)-\frac{1}{2}g_{bc,a}\dot{x}^b\dot{x}^c=f(\lambda)g_{ab}\dot{x}^b$

where $f$ a general nonzero function. I had been trying to do this by varying the action

$S=\int L \textrm{d}\lambda$ with Lagrangian $L=\frac{\textrm{d}s}{\textrm{d}\lambda}=\sqrt{g_{ab}\dot{x}^a\dot{x}^b}$.

The standard technique is use as a Lagrangian $L'=L^2$ instead, and claim that this produces the same results as we would have got with $L$. I’d always accepted this as gospel, but a simple calculation shows that we need an additional assumption for this to be true.

Indeed the Euler-Lagrange equations for $L'$ give

$\frac{\textrm{d}}{\textrm{d}\lambda}(2L\frac{\partial L}{\partial\dot{x}^a})=2L\frac{\partial L}{\partial x^a}$

To regain the Euler-Lagrange equations for $L$ we want to cancel our $2L$ factors. Clearly a sufficient condition for this is that $L$ is constant. But by the definition of $L$ this means that $\frac{\textrm{d}s}{\textrm{d}\lambda}$ is constant, which precisely says that $\lambda$ is an affine parameter.

Hence, by the use of this method we lose the generality needed to obtain equation (*).

Nevertheless it is easy to derive (*) from the affine geodesic equation (**)

$\frac{\textrm{d}}{\textrm{d}\lambda}(g_{ab}\dot{x}^b)-\frac{1}{2}g_{bc,a}\dot{x}^b\dot{x}^c=0$

Indeed let $\mu$ be a general parameter and write $\lambda = \lambda(\mu)$. Then rewriting the geodesic equation (**) in terms of $\mu$ using the chain rule yields (*) with $f(\mu)=\frac{\lambda ''}{\lambda '}$ (try it).

Alternatively you can reach a manifestly reparameterisation invariant version of (*) by directly varying our original action $S$. It’s not pretty though!