How to understand Einstein’s relativity without math

120 years ago, a revolution took place in physics that — to an outsider — might seem like an inconsequential matter. 120 years ago, Einstein put forth his Special theory of Relativity, asserting that neither space nor time were absolute quantities, but rather the answers you’d get for measuring distances, positions, and durations would be dependent on your location and relative motion. The only absolute, Einstein contended, was the speed of light in a vacuum. This was indeed a revolutionary statement, but the formulas for working out how distances and durations changed in a velocity-dependent way, especially as you approached the speed of light, had already been worked out over a decade prior: the Lorentz transformations.

And yet, Einstein’s key insights, and the profundity of Special Relativity, were lauded and marveled at by even the physicists who had discovered the transformations previously. Even today, most laypersons only have a vague notion that “distances and times are relative” instead of absolute, not a solid understanding of how or why that is. Most people who study physics can only refer to the Lorentz transformation’s equations and plug numbers in, not explain why times dilate and lengths contract, particularly near the speed of light. And many people, when it comes to relativity, don’t appreciate the difference between Einsteinian relativity and the main principle of relativity that goes back much farther: to the time of Galileo.

And yet, these three simple principles, if you can keep track of and remember them, will give you a solid understanding of Einstein’s relativity without the need for math at all. Here’s what everyone should remember.

Different frames of reference, including different positions and motions, would see different laws of physics (and would disagree on reality) if a theory is not relativistically invariant. The fact that we have a symmetry under ‘boosts,’ or velocity transformations, tells us we have a conserved quantity: linear momentum. The fact that a theory is invariant under any sort of coordinate or velocity transformation is known as Lorentz invariance, and any Lorentz invariant symmetry conserves CPT symmetry. This notion of invariance under constant motion dates all the way back to the time of Galileo.

1.) Relativity didn’t start with Einstein, but goes back some 300 years before him.

Today, the concept of relativity is synonymous with Einstein: Special Relativity for the motions of objects close to the speed of light in otherwise flat, uncurved space, and General Relativity for the motions of objects at any speed in gravitationally bent/curved space. We normally think of relativity as telling us that there is:

• no such thing as an absolute distance between two objects or two points,
• no such thing as an absolute duration, or time, between two events,
• and no such thing as a “preferred reference frame,” where one observer can claim that their perspective is “more correct” than any other.

All of these things are indeed true in Einstein’s relativity, but the concept of relativity — which specifically concerns itself with the existence of an absolute reference frame — is much, much older than an Einsteinian concoction.

Instead, relativity goes back nearly 400 years before the present (and nearly 300 years before Einstein), when Galileo was talking about a far simpler issue: the detectability of motion. First written down in 1632, Galileo was thinking about a passenger aboard a ship in the lower decks. Imagine that there were no windows, no portal, no vision to the outside world at all. Galileo imagined two scenarios, and asked the reader to consider what detectable differences there would be between:

• someone in the closed room in a ship that was at rest with respect to the ocean, the shore, or some other point-of-reference,
• and someone in the closed room in a ship that was in constant, unchanging motion with respect to the ocean, the shore, or some other point-of-reference.

Are these men on a train that’s stationary and at rest? Or is the train in constant motion relative to the outside world? If not for the sounds and vibrations of the train along the track, and the cues of the outside world seen through the windows, there would be no way for them to know, and no experiment they could perform inside the train car that would tell them. This is the key behind Galilean relativity.

Galileo’s great realization, encapsulated in the principle of relativity, is that there would be no detectable difference at all. Any experiment you performed, from jumping up-and-down to releasing an object in free-fall to launching a projectile to rolling a ball up-or-down a ramp, would yield identical experimental results under these two scenarios. The principle of relativity would later show up in Newton’s work as his first law of motion, where an object at rest would remain at rest and an object in motion would remain in constant motion, and would do so forever, unless and until either of them was acted upon by an outside (net) force. Galileo wrote precisely about this case in his 1632 book, Dialogue Concerning the Two Chief World Systems, where he stated (English translation follows):

“Shut yourself up with some friend in the main cabin belowdecks on some large ship… [and] a person belowdecks on a smoothly sailing ship cannot hope to determine whether the ship is in motion or, if so, what is its speed.”

This was the birth of relativity: the notion that someone in motion has no physical evidence of their motion except through comparison with some measurable reference to the outside world. To Galileo, something like “position” or “distance” or “velocity” is not a quantity that can be absolutely defined, where all observers — including ones on the ship, ones on the shore, ones in a canoe, etc. — will measure different values. The laws of physics that govern mechanics remains the same between observers, but the exact results for these quantities will depend on each observer’s relative motion to the system they’re observing. That was the original version of relativity, and it’s still basically correct even today.

When you move a magnet into (or out of) a loop or coil of wire, it causes the field to change around the conductor, which causes a force on charged particles and induces their motion, creating a current. The phenomena are very different if the magnet is stationary and the coil is moved, but the currents generated are the same. This was the jumping-off point for the principle of relativity.

2.) Regardless of an observer’s motion or frame of reference, the physics should remain the same.

This seems like it’s not a new statement, as Galileo surely realized that this was the case for the motions of objects under the influence of Earth’s gravity: the science of classical mechanics. But back in the time of Galileo, the phenomena of electricity and magnetism were only understood in the most rudimentary of fashions; by the 19th century, they had been investigated much more thoroughly.

Electricity and magnetism, at a fundamental level, relies on the physics of electric charges: stationary electric charges generate electric fields, while electric charges in motion generate electric currents, which in turn generate magnetic fields.

Moreover, if you change the magnetic field within a loop of wire, there will be an electric current (and hence, moving electric charges) generated within the wire.

When you take these two pieces of information together, that leads to some profound implications. One of them is this: if you move relative to a stationary electric charge, you won’t just see its electric field (and it will have one), but because it’s in motion relative to you, it must create a magnetic field, too.

If you have a stationary electric charge (top left) and you move relative to it (top right), you’ll see the particle’s one-symmetric electric field change and become compressed along the direction of motion, and also a magnetic field will appear. If you start with a particle in motion (bottom left), it will exhibit both an electric field and also will generate a magnetic field from the electric current. If you boost yourself to move along with the particle (bottom right), the magnetic field will disappear and the electric field will become spherically symmetric once again.

If you have a charged particle that moves relative to the stationary charge, it’s going to be bent by the electromagnetic forces that act on it. However, in one frame of reference, it’s electrostatic forces only that cause the deflection, and in another frame of reference, it’s both electric forces and magnetic forces that cause the deflection. You can work this out for yourself, quantitatively, if you like; many physicists did precisely this in the 19th century.

The kicker is this: when you do work it out, what you find is that the electric field generated by the charged particle at rest and the electric field generated by the charged particle in the frame of reference where it’s moving cannot be the same electric field! According to the laws of electromagnetism, if you calculate:

• the effects of the electric field on the particle in the “at rest” case,
• and the effects of the magnetic field on the particle in the “moving” case,

then you’ll find that if you want the principle of relativity to be true — i.e., you want the physics of how the charged particle deflects to be equivalent in both scenarios — then the electric field in the “moving” case cannot be the same as the electric field in the “at rest” case. Instead, the electric field needs to contract along the direction of motion by a specific amount: by the amount dictated by the Lorentz transformations. If you make that transformation, then the trajectory of the charged particle becomes equivalent between the two cases. It was by considering electromagnetism, and not classical mechanics, that the first hints of relativity actually showed up.

Light is nothing more than an electromagnetic wave, with in-phase oscillating electric and magnetic fields perpendicular to the direction of light’s propagation. The shorter the wavelength, the more energetic the photon, but the more susceptible it is to changes in the speed of light through a medium.

3.) The key to unlocking relativity, then, came by considering the one thing that was invariant in electromagnetism: light.

Light, as Maxwell first realized, is an electromagnetic wave. It propagates at the speed of light, is composed of oscillating, perpendicular, in-phase electric and magnetic fields, and carries energy. Einstein — at least, according to legend — imagined trying to follow this light wave as quickly as possible, watching the electric and magnetic fields in front of him oscillate in the process. He imagined going faster and faster, until it seemed like the oscillating fields were barely ahead of him, and that’s when he had a spectacular realization.

There was no situation where this was possible. You couldn’t ride with a light wave; you couldn’t follow a light wave; you couldn’t do anything that would make that light appear to travel at any speed other than the speed of light.

This was bolstered by an experiment (or, more accurately, a series of experiments) performed in the 1880s: the Michelson-Morley experiment. By noting that the Earth moves at about 30 km/s relative to the Sun, the scientists created an interferometer setup where light was:

• split into two beams,
• of identical path lengths,
• with mirrors at the end,
• with one beam sent along Earth’s direction-of-motion,
• and one beam sent perpendicular to Earth’s direction of motion,
• and then the two beams were recombined, and an interference pattern was searched for.

The entire interferometer could be rotated at a variety of angles, and if there was a stationary medium for light to travel through (what most assumed the “aether” was), this experiment would reveal its effects.

If you split light into two perpendicular components and bring them back together, they will produce an interference pattern. If there’s a medium that light is traveling through, the interference pattern should depend on how your apparatus is oriented relative to that motion. If the speed of light is a constant to all observers, however (a contradiction of Newton’s predictions), then light will arrive from even mutually perpendicular directions at the eventual detector simultaneously.

But instead, the experiment only saw a null result: arguably the most famous and impactful null result in history. It was with all of these pieces of data — and all of these thoughts — in combination that Einstein had his big breakthrough.

• Space wasn’t absolute, but relative.
• Time wasn’t absolute, but relative.
• Speeds weren’t absolute, but relative.
• Electric and magnetic fields weren’t absolute, but rather were velocity dependent.
• But one thing did remain the same for any and all observers: the speed of light.

That was it; that was the big realization that led to relativity: the notion that the speed of light was the one invariant, or thing that remained the same for all observers, in the Universe.

If you realized this, then it suddenly became possible to derive the Lorentz transformations, for both the phenomenon of time dilation and also for that of length contraction, just by imagining how light traveled. Let’s say you have a clock that works by reflecting a photon (or light wave) up-and-down between two mirrors. If you are at rest, you can measure time by counting how many oscillations (or trips up-and-down) the photon makes. But if you’re watching someone in motion, and in particular, in fast, relativistic, near-the-speed-of-light motion, you’ll see their “light clock” experiences time passing at a very different rate.

A “light clock” will appear to run differently for observers moving at different relative speeds, but this is due to the constancy of the speed of light. Einstein’s law of special relativity governs how these time and distance transformations take place between different observers. However, each individual observer will see time pass at the same rate as long as they remain in their own reference frame: one second-per-second, even though when they bring their clocks together after the experiment, they’ll find that they no longer agree.

What is the rate that time ticks by for them?

Well, as you can see, above, the light clock that’s in relative motion has a slower “ticking rate” than the light clock that’s at rest. Because you know:

• what the speed of light is,
• that the speed of light is invariant,
• and that we understand the mathematics of geometry in general, and of right triangles in particular,

we can calculate how fast the moving light clock ticks by relative to the stationary one.

As you might have anticipated, the time dilates by exactly the amount predicted by the Lorentz transformations. If time is dilating, however, and velocity is a change in position (or distance) over a change in time, then position has to change relativistically as well. However, because it’s the speed of light that’s invariant, not the “rate of ticking” or the distance between two points, distance has to change in the inverse way that time changes: whereas times dilate, or get longer, for the observer that appears to be in relative motion, lengths need to contract, or appear shorter, for an object in motion relative to an object at rest.

One revolutionary aspect of relativistic motion, put forth by Einstein but previously built up by Lorentz, FitzGerald, and others, is that rapidly moving objects appear to contract in space and dilate in time. The faster you move relative to someone at rest, the greater your lengths appear to be contracted, while the more time appears to dilate for the outside world. This picture, of relativistic mechanics, replaced the old Newtonian view of classical mechanics, but also carries tremendous implications for theories that aren’t relativistically invariant, like Newtonian gravity.

This is perhaps the simplest way to understand Einstein’s relativity: not necessarily the exact way Einstein himself first made sense of it, but a way that, with more than 100 years of hindsight, we can look back and try to make it as intuitive as possible. There are many equivalent, more mathematically intensive ways to look at this.

• We can weave space and time into a fabric, as Einstein’s former teacher, Hermann Minkowski, would do in 1908, and recognize that it isn’t just the speed of light that’s invariant, but also a four-dimensional quantity known as the Einstein (or spacetime) interval.
• We can recognize that while we have three dimensions of space and one dimension of time, the “time dimension” is fundamentally different because of the minus sign that shows up in the square of the quantity, meaning that times lengthen while distances contract near the speed of light. (This is why, for the mathematically inclined, Special Relativity often leverages hyperbolic coordinates.)
• Or, we can simply calculate what an observer experiences as they journey, at rapid speeds, to an interstellar star system: from the point-of-view of someone who remains on Earth and from the point-of-view of someone who’s on board the journeying ship.

If we consider this last case, it’s easy to understand why, if times dilate, lengths also need to contract.

This moving, zipping star field appears to depict an ultra-relativistic motion through space, extremely close to the speed of light. Under the laws of relativity, you neither reach nor exceed the speed of light if you’re made of matter. To a stationary observer, clocks on the spaceship moving at this speed appear to run slow (times dilate), while to someone on the spaceship, the distances along their direction-of-motion appear to contract (length contraction).

From the perspective of a stationary observer on Earth, the nearest singlet Sun-like star is Tau Ceti, about 12 light-years away. If a spaceship traveled towards Tau Ceti at 99.65% the speed of light, you’d watch the time on the spaceship’s clock tick by much more slowly than your own clock ticked by: for every 13 seconds that passed for you, just one second would appear to elapse for the spaceship. From your perspective, they would traverse 12 light-years of distance in just over 12 years, but they themselves would only age by approximately one year. It’s a remarkable prediction of relativity.

But for someone aboard the spaceship, how is that possible? How could they traverse a distance of 12 light-years and only experience one year of time elapsing? Wouldn’t that imply they were moving faster-than-light?

Only if you ignored length contraction. For someone in motion, it’s the distances along that direction-of-motion that contract. At 99.7% the speed of light, those 12 light-years will contract to appear to be just about 0.92 light-years, which they can then reach in just under one year of travel at relative speeds of 99.7% the speed of light. The key to remember is this: lengths contract and times dilate near the speed of light, but that the speed of light itself is invariant, and the same for all observers, regardless of their relative motion to absolutely anything. Even without math, just thinking about the physics of what’s going on can lead you to develop something that eluded most of Einstein’s peers: an intuition for how the Universe behaves as you near the speed of light!