The Hitchhiker's Guide to Relativity

Source: NASA

Part 1: Special Relativity

When looking at a painting, the first thing you see is the image. Whether big or small, beautiful or grotesque, antique or modern, you let the effect it has wash over you. But on closer inspection, you can see the complex pattern of the brushstrokes, the meticulous planning, and the hard work that went into bringing the inspiration to life. Astronomy and its physics are kind of like that. Its muses – the stars, planets, and the universe itself – are of exquisite beauty but also of bewildering complexity; capturing their essence and bringing it down to Earth requires a special kind of inspiration, and endless math – the brushstrokes of physics. One such masterpiece is Einstein’s Theory of Relativity.

Though elaborate a topic enough to fuel several degrees, a good introduction to it is thinking about how everything is constantly in motion. Earth is hurtling through space at 1,670 km/h, the Sun at 720,000 km/h, and the Milky Way is cruising at a casual 2.1 million km/hr. How can we understand anything at all with this madness? Before we get into the nitty-gritty, we must first realize that, as its name suggests, relativity and its applications depend on the (relative) motions of various objects and the observer (standpoint from where we are looking at these motions from). This may be difficult to wrap your head around, like a lot of modern physics (or just physics in general), but it just requires abstract thought and imagination to see the logic.

Imagination and creativity? In math? It seems strange, but the reason Albert Einstein was so good at what he did was because he saw things in his mind’s eye – through so-called thought experiments – and dealt with the math from there. In 1905, he turned the world of physics on his head when he published his paper On the Electrodynamics of Moving Bodies, where he described what we now know as special relativity. It stemmed from a thought experiment he came up with when he was 16 years old. In it, he imagined what would happen if he chased and caught up with a beam of light.

As Galileo described in his own principle of relativity (which, while illustrating the following point well, only holds true at relative speeds that are much less than the speed of light), motion is relative in all inertial frames. What this means is that when objects are either at rest, or moving at a constant velocity (with no forces acting on it) – these are inertial frames – its movements are relative to each other. For example, as described here, someone sitting in one of two parallel trains which are moving at the same speed would be able to have a conversation in sign language with someone sitting opposite in the other train; the trains are not moving relative to each other. Similarly, picture a moving train that is not accelerating i.e., is at a constant speed. One would not be able to tell it was moving without looking outside, because everything inside looks normal (assuming it is not a bumpy ride). But Galileo only got part of the answer.

Picturing himself running at pace with a beam of light, Einstein figured he would be able to see the light in static form. According to the physics at the time, he should have been able to; it was thought that the speed of light depended on its source (emission theory), and by catching up with it, one could see its constituents. But now we know that this is impossible; ‘Einstein hypothesized that, if something could travel faster than the speed of light, it would break fundamental physical laws by being able to observe, relatively speaking, a stationary electromagnetic wave’.

In a nutshell, through a mixture of intuition, lack of experimental proof for the emission theory (but proof for a constant speed of light), and some scary math in the form of the Maxwell equations (a complex set of equations relating to electromagnetism), Einstein realized how light behaves differently than anything else in the universe: its speed in a vacuum is always constant. No matter how fast the source of the light is moving, where it is going, or how big it is, the speed of light will always be around 300,000 km/s relative to the observer.

Einstein also theorized that ‘the laws of physics take the same form in all inertial frames of reference’; in other words, not only mechanics (as Galileo said), but all of physics is relative. Combine this and the speed of light being constant, and you’ve got the rules of special relativity down.

Well, not completely. Because while the two precedents described above sound simple, their derivation but also their consequences are pretty baffling. The fact that the speed of light is constant opens up a whole new can of worms in physics. A good example is used here: if you are on a train and walk down the aisle in the same direction that the train is going, the trees and buildings outside pass by just a little quicker than if you were standing still. If the train was headed straight for the sun, by the same logic, its light should come at you a little quicker if you were walking towards it. But the speed of light is constant, which meant something else… wasn’t. This really bothered Einstein, until, with the help of his friend Michele Besso, it came to him.

It was time. Time was relative. In another thought experiment, Einstein pictured yet another train, this time with someone sitting inside it, and another person standing on a platform (inertial frames of reference). As the midsection of the train passes by the platform, two lightning bolts strike the train: one at the front, and one at the back. To the person on the platform, they happen simultaneously. But to the person sitting inside, they happen in quick succession; the light from the bolt at the front of the train reaches the passenger a little bit quicker because the train is moving in that direction, leaving more distance that the light from the posterior bolt needs to cover to catch up. But both observers are right, despite experiencing the event differently. Therefore, Einstein figured, simultaneity itself is relative. Something that happens at the same time for me could happen at two different times for you.

And it gets crazier. Going back to the example of the train hurtling towards the sun (yes, relativity loves trains): we had established that as the train speeds towards the star, the speed of light cannot speed up; it’s constant. So, it is time that has to slow down in order for light to remain constant. The sun is just an example for visualization; basically, the faster you go, the more time slows down for you. This is known as time dilation.

Here's one more of Einstein’s handy thought experiments to help picture this concept. No trains this time, I promise. Instead, picture two small mirrors facing each other – one facing upwards, one downwards – and a light pulse moving vertically between them. This will serve as our clock – as the speed of light is constant, the time it takes for the light to bounce vertically between the mirrors remains constant also, and that is our unit of time measurement. But say we take this clock with us on an extremely fast spaceship, and someone is somehow able to see the clock inside the ship as we go.

As the light pulse leaves the lower mirror, the rocket – and the clock – move forward. As explained here, ‘this means that for the observer stood outside the rocket looking in, the light beam will be hitting the mirror further…than the point it was emitted from’. This results in a longer, stretched diagonal route that the light takes as the whole contraption moves with the rocket. Because the speed of light is constant, it means the light takes a longer time to reach the opposite mirror. For the person in the spacecraft, it looks normal, because they and the clock are moving at the same speed. But to someone on Earth (who can peer inside the spacecraft and see the clock), it is possible to see time pass more slowly on the spaceship.

Scarily, if the rocket took a trip of a few minutes at extreme speeds, by this logic, much more time would have passed back on Earth. But this also means that if we were able to build a ship capable of going at relativistic speeds (or really, really fast), due to time dilation, the people on the ship would experience time much more slowly. So, while the rest of us on Earth must wait decades for the astronauts to reach their destination, to them, it may only take, say, one year. We do not (yet) have the technology to go that quickly, but on the ISS, astronauts age about 0.005 seconds less compared to those on Earth. However, gravity also has an effect on time dilation, and that’s a whole other nut to crack.

But it’s not only time that is relative. Another variable that does not stay constant at significant speeds is space, or distance; at these speeds, the length of an object looks different depending on where you are looking at it from. The phenomenon is known as length contraction. This seems crazy, and it is really only observable at relativistic speeds. But mathematically, it is quite logical. Despite time dilation, both observers still witness the same relative speed, which you can get by dividing distance by elapsed time. Because the time is different for each observer, the distance must be different too. Einstein’s realization that space and time were interwoven led to him declaring them as part of a four-dimensional structure called spacetime, in which all objects in the universe are suspended: three dimensions of space, like the world around us, with time being the fourth.

You’ve probably heard that time stops at the speed of light (for outside observers watching); that is due to time dilation, which increases with speed. This also explains length contraction, because, for someone watching from the outside, light takes a shorter amount of time to reach across the object and the fast-travelling object would appear squished (just like the forward lightning bolt’s light took less time to reach the passenger on the moving train than the other). When travelling at the speed of light, the light needs no time to travel across the object, so its observed length is zero. If you were the one travelling at that speed, your time and proportions would seem normal, and everything else would be different. It’s all a matter of perspective; it’s relative.

But this is all extremely hypothetical, mainly because unless one is a massless particle like a photon, going at the speed of light is impossible. Thanks to Einstein’s famous equation e=mc2, we know that energy is proportionate to mass times the speed of light squared. As explained here, we need an incredible amount of energy to accelerate to even a fraction of the speed of light, so it would also take more mass. It would take an infinite amount of energy to accelerate to the speed of light, and using the equation, that would also take an infinite amount of mass. So unfortunately, at the moment it seems like that is the speed limit.

Relativity is pretty much the cornerstone of modern physics, and mind-bending as it is, this is just the beginning; we haven’t even started on general relativity. Even in this summary, the dizzying equations that play a big part in it are left out. The concepts, however, are truly beautiful, and once you understand those, it leads to not only a better grasp on astrophysics and rocketry, but also the whole universe. The stripped-down physics may not have an effect in our daily lives, but there’s something about realizing that everything is relative that just opens your mind. Something as simple as a thunderstorm, where one person experiences the flash and thunder at the same time and another with an interval between them, can illustrate this concept. While still restrained by the laws of physics, it allows for individuality within them; it makes you appreciate that everyone experiences the universe in a slightly different way, just like looking at a painting.

Part 2: General Relativity

Einstein’s 1905 theory of special relativity alone allows one to see the artwork of the universe from a new perspective. Before we go on, here’s a short recap: special relativity stated that the speed of light is constant in a vacuum no matter where you look at it from, and that the rules of physics are unchanged from any vantage point that is not accelerating (inertial frame). It involved the formula e=mc2, which casually stated that matter and energy are somewhat interchangeable. Its consequences are vast, including that time itself is relative. It also suggested the idea of a four-dimensional fabric or structure called space-time, which held all of the objects in the universe. But with all this flashiness, it did not account for the mammoth variable that is gravity. That’s where general relativity comes in.

General relativity is all about gravity. Before Einstein came out with it in 1915, Isaac Newton’s Law of Universal Gravitation was used as standard, and still is quite an accurate generalization of the topic. It states that ‘any particle of matter in the universe attracts any other with a force varying directly as the product of the masses and inversely as the square of the distance between them’. Essentially, it explains that gravity is a force; an attraction. It’s a mouthful, but given the complexity of the subject, it works remarkably well. But even Newton himself was bothered by how little was known about this force, and there are a few small discrepancies in the law; for example, the orbit of Mercury has anomalies that could not be explained by Newton’s law, and it also made faster-than-light travel out to be possible. So, while Newton’s law is impressively accurate for the most part, for extreme situations (such as black holes and close-to light speeds) and miniscule details, Einstein’s theory is needed.

In a tiny nutshell, Einstein’s general relativity states that gravity is the warping, or curving, of spacetime (the four-dimensional structure mentioned in special relativity: three dimensions of space, and one of time), and tells us how to measure it. Without any mass in it, spacetime would be flat. But as soon as objects with mass – like planets, stars, galaxies, etc – join the game, they mold, or warp, the spacetime landscape around them. This, in turn, affects the motions of other objects: gravity.

This warping makes gravity out to be not a force, as Newton said, but a ‘geometric property’. As shown by NASA, spacetime, while quite literally beyond our scope of vision, can be pictured as a bedsheet stretched out between a few people. Now, place a billiard ball in the center. The bedsheet curves downwards. If you place a marble onto the edge, it will roll down the slope towards the ball. That is gravity. Now imagine a bowling ball in the center, and placing a marble onto the sheet. The slope would become much steeper and take up more of the bedsheet, and the marble would roll down much faster and from a further distance. This model does not account for the complexities of general relativity, but summarizes the basic principle of spacetime.

Einstein wanted to combine gravity (or Newton’s law, which was what was accepted at the time) with his own special relativity. In what is called the Equivalence Principle, Einstein realized that with a lack of acceleration comes a feeling of weightlessness. Einstein used a thought experiment for this, shown here: if you are inside an elevator and you find yourself to be floating, unless you look outside, you would not be able to tell whether that elevator was floating in space (without gravitational influence), or falling downwards at a constant velocity. This is because the elevator and everything in it is ‘falling’ at the exact same rate. To illustrate: aboard the ISS, Earth still has significant gravitational influence, but the station is constantly falling ‘around’ the Earth, which results in the weightlessness. Because it is impossible to tell the difference between true weightlessness and a free-fall situation in isolation, Einstein showed that free-fall, by definition, is an inertial frame (a state with no acceleration).

Einstein also realized the flipside of this. Imagine you are standing in a locked room with no windows. You realize you must be on Earth because you feel the planet’s familiar gravitational pull of 9.81 m/s2. But are you? It turns out, the room could be inside a rocket that is accelerating at 9.81 meters per second, and you would feel absolutely no difference to Earth’s gravity. You could play basketball and not feel a thing. So, sort of like free-fall/no acceleration feeling the same as zero gravity, the presence of acceleration can feel the same as actual gravity, and it is impossible to tell the difference. In either situation, you would be in a non-inertial frame (a state with acceleration). This is the equivalence principle, and it basically equates gravity with acceleration, and a lack of gravity with a lack of acceleration (at least locally).

Let’s talk about mass for a second. As explained here, an object has two types of mass. The first is inertial mass: how much an object can resist acceleration (for example, a boulder would have a high inertial mass). The second is gravitational mass: the strength of gravity generated by an object. Considered a bizarre phenomenon in physics, these two types of mass are always the same for a given object; but as explained above by Einstein and the equivalence principle, which describes acceleration as comparable to gravity, it makes sense. Therefore, mass remains the same whether we are dealing with inertia or gravity. This suggests gravity is a pseudo-force instead of a ‘real’ force, which supports the theory of curved spacetime and its geometrics.

But the equivalence principle is not the whole story; it cannot explain the tidal effect, which can be seen in the following example: picture two basketballs in an elevator that is free falling towards the Earth. They will not fall in parallel lines, but rather form a wedge shape; they come closer to each other on their way down (as shown here). This contradicts the generally accepted idea that objects will travel at a constant velocity – straight line – unless something acts on it. As shown by the equivalence principle, falling towards the Earth (for example) and traveling at a constant velocity are comparable (in a small confined space) in that we consider them to have no external forces acting on them. But we’ve just seen that two parallel objects in free fall do not travel in a straight line. This shows that the straight lines as we know them themselves are curved.

Going back to spacetime, in the absence of any gravity at all, particles (let’s ignore their own gravity) will move in a straight line forever (as described by Newton’s first law). But objects with mass (and we now know inertial and gravitational masses are equivalent) can curve spacetime towards them, like the bedsheet. To understand this more fully, let’s talk about something called geodesics.

Geodesics are, in the most basic terms, the shortest paths an object (that is not accelerating) can take between two points in spacetime. As explained here, if the spacetime is completely flat, the path will be a straight line, but if it curves around something with a large mass, like a planet, an object will follow that curve. While there’s a lot more to it than that, this explains tidal forces in that each object will take the shortest path with regards to the curvature of spacetime. Those basketballs are, individually, taking the shortest path to the Earth’s center of gravity – its core – and that path happens to be curved. Seen this way, gravity is technically not keeping us standing on the surface of the Earth; instead, Earth’s crust is holding us up as gravity demands we fall to its core.

So according to all the above, how do objects stay in orbit? An object’s geodesic, or path, is essentially a combination between its velocity and gravitational forces acting on it, or the curvature of spacetime. When an object’s velocity is perpendicular to the gravitational source and it is fast enough, its geodesic will be an orbit around that source. Light, due to its extreme velocity (speed of light) is minimally affected by gravity, which is why there isn’t any of it orbiting Earth. However, look at a supermassive black hole with super strong gravity, and you’ll see an accretion disk of light orbiting it.

While we’re at it, we can’t forget that the fourth dimension of spacetime is time, and this has some interesting implications. If you’re now familiar with special relativity, you will recall the effect it has on time dilation. But general relativity and its incorporation of gravity adds another layer to that. As spacetime curves and flexes (like the bedsheet), so does time. That means that time runs more slowly closer to a gravitational source; as part of the fabric of spacetime, it is more ‘stretched’ closer to the source. As mentioned here, the Earth’s core is about 2.5 years ‘younger’ than its surface, as its gravitational pull is slightly stronger there. Conversely, time passes faster the further away one is from the source. This was predicted by Einstein and has been proven experimentally since.

An extreme and rather terrifying example of the effect would be watching someone falling into a black hole: a region of unthinkably strong gravity. From the outsider’s perspective, time passes much, much faster compared to the goner. This means that while the person falling would meet a swift and hopefully painless end, to the outsider, the extreme time dilation makes it look like they are permanently falling. If you’ve ever seen Interstellar, the same principle is shown there. As the crew find themselves close to a black hole for what seems like just a little while, the years whizz by back on Earth.

If that isn’t cool enough, with general relativity came a host of new predictions, resulting in discoveries, about gravity. One is gravitational redshift (more thorough explanation here). Redshift occurs as a result of the Doppler effect, in which light waves are either squished (when the source is moving towards you: blueshift) or stretched (source moving away: redshift). As gravity is comparable to acceleration, when shining a beam at the ceiling, you can think of the ceiling as accelerating away from the light source, causing the emitted light waves to be stretched, resulting in redshift.

Another discovery is gravitational lensing. We have already learned that light bends around massive objects, and gravitational lensing is a consequence of that. A big galaxy, for example, can have enough mass to curve surrounding spacetime enough so that the light around it gets caught in the curvature and appears bent to us. The effect is directly observable through telescope images, though relatively rare. This can even allow us to see objects, such as galaxies, that are actually located directly behind the massive object in the forefront. In fact, were we to place a telescope at the right point in space, we could use the Sun as a gravitational lens to be able to see certain objects such as exoplanets better.

One last example of a neat discovery is gravitational waves, predicted by relativity but first detected in 1974. They are formed by incredibly strong gravitational events, such as a supernova, or a black hole collision. This will cause ripples in spacetime which then travel across the universe, like waves in the sea. They are too complex a topic to cover here in full, but rest assured that although the events that cause them are cataclysmic, by the time they reach us, the waves are so tiny that scientists need extremely sensitive instruments to pick them up, and we humans don’t even feel them. What is most mind-blowing about all of this, to me, is that for the most part, these discoveries were abstract predictions made by Einstein. Sure, there’s the perplexing math to back it up, but discoveries are still being made that prove him right.

It’s frustrating to know that this is only the briefest of overviews on general relativity; to include every detail and consequence would turn this blog post into a novel, and that’s not even considering the relentless math. But this just emphasizes the marvel of the human mind that can take the complexity of the universe and frame it in such an elegant way that astronomers and physicists are still milking it today. That’s not to say it was a walk in the (Leopold)park; it is the culmination of years of work, frustration, collaboration, and question marks. But what really makes it stand out is that so much of it is based on the musings, imagination, and inspiration of its creator. Its artistry was born from pure curiosity and his desire to explain the universe around him. Relativity is known as the most beautiful theory ever created, and as with all great art – be it the music of Bach or the paintings of Da Vinci – even if you can’t see or understand every note or brushstroke that went into it, its brilliance is incontestable.